U  " 

TOTHE 


WESTERN 


CONTAINING 


SOLUTION  OF  ALL  THt  EXAMPLES  AND 
QUESTIONS 


.  E  REFERENCE  TO  THE  PAGES  WHERE  THEY    STAND. 


TO  WHICH  IS  ADDED, 


<!lsefta 


ESIGNED  CHIEFLT  TO  FACILITATE  THE  LABOUR  OF  TEACHERS; 

AND  ASSIST  SUCH  AS  HAVE  NOT  THE  OPPORTUNITY 

OF  A  TUTOR'S  AID. 


BY  JOHff  ARMSTRONG. 

Ufc. 


SThtrtr  SBUttion,  rrttfsrt  sntJ  cocrmetr. 


PITTSBURGH: 

RINTED  AND  PUBLISHED  BY  JOHNSTON  fc  STOCKTON, 
MAliKET    STREET. 


1831. 

JOSEPH  B 


WESTERN  DISTRICT.  OF  PENNSYLVANIA,  to  wit: 

•  BE 'IT  'REMEMBERED,  That  on  the  twenty-fourth 
^ay  of  April,  in  the  forty-eighth  year  of  the  Independence  of 
g  the  United  States  of  America,  A.  D.  1S24,  Eichbaum  &f  John- 
4fe  ston>  of  the  said  district,  have  deposited  in  this  office  the  title 
of  a  book,  the  right  whereof  they  claim  as  proprietors  in  the  words  follow- 
ing, to  wit: 

"A  Key  to  the  Western  Calculator,  containing  the  solution  of  all 
the  examples  and  questions  for  exercise,  with  reference  to  the  page* 
where  they  stand. — To  which  is  added,  some  Useful  Rules.  Design- 
ed chiefly  to  facilitate  the  labour  of  teachers;  and  assist  such  as  have 
not  the  opportunity  of  a  tutor's  aid.  By  JOHN  ARMSTRONG." 

In  conformity  to  the  act  of  the  Congress  of  the  United  States,  entitled, 
"  An  act  for  the  encouragement  of  learning,  by  securing  the  copies  of 
maps,  charts  and  books,  to  the  authors  and  proprietors  of  such  copies, 

during  the  time  therein  mentioned." And  also  to  the  act,  entitled,  "Jin 

act  supplementary  to  an  act,  entitled,  "Jin  act  for  the  encouragement  of 
learning,  by  securing  tJie  copies  of  maps,  charts  and  books,  to  the  author* 
and  proprietors  of  such  copies,  during  the  time  therein  mentioned?  and 
extending  the  benefits  thereof  to  the  arts  of  designing,  engraving,  find 
etc/ting  historical  and  other  prints.13 

Wm.  WALKER, 
Clerk  of  the.  Wettcrn  District  of  Pcimsylvatua. 

CAJORI 


Numeration 

Addition        -        -        -        -  ib. 

Subtraction         ...  6 

Multiplication  7 

Division  £ 

Federal  Money         -  12 

Compound  Addition    -        -  1£ 

Compound  Subtraction'    -    -  14 

Compound  Multiplication  1C 

Compound  Division      -        -  18 
Reduction 

Addition  of  Decimals         -  3C 

Subtraction  of  Decimals  31 

Multiplication  of  Decimals  ib. 
Division  of  Decimals 

Reduction  of  Decimals        -  33 

,  Single  Rule  of  Three  Direct  37 

'Single  Rule  of  Three  Inverse  41 

Double  Rule  of  Three        -  47 

Practice        -        -        -        -  53 

Tare  and  Tret            -        -  C>0 

Simple  Interest        -        -  62 

Compound  Interest      -        -  68 

Insurance        -        -          -  ib. 
Commission 

Brokage        -        -        -  ib. 

Buying  and  Selling  Stocks  70 

Rebate  or  Discount  ib. 

Bank  Discount         -         -  71 
Equation  of  Payments 

Single  Fellowship        -        -  ib. 

Compound  Fellowship      -  74 

Profit  and  Loss        -        -  75 


Page. 

Barter  ....  77 
Exchange  ...  78 

Alligation  80 

Reduction  of  Vulgar  Fractions  82 
Addition  of  Vulgar  Fractions  86 
Subtraction  of  Vulgar  Fractions  88 
Multiplication  of  Vulgar  Frac.  ib. 
Division  of  Vulgar  Fractions  89 
Rule  of  Three  in  Vulgar  Frac.  ib. 
Inverse  Proportion  -  -  90 
Involution  ...  ib. 
The  Square  Root  -  -  91 
The  Cube  Root  -  -  96 
Arithmetical  Progression  -  102 
Geometrical  Progression  103 
Single  Position  106 

Double  Position  -  -  107 
Permutation  -  -  111 

Combination  -  -  -  112 
Duodecimals  ib. 

The  Carpenters' Rule  -  115 
Measuring  of  Boards,  &c.  117 
Carpenters'  &  Joiners'  Work  118 


Bricklayers'  Work 
69  Masons'  Work 

Plasterers'  Work 

Pavers'  Work    - 

Painters'  Work 

Glaziers'  Work 
72  Measurement  of  Ground 


Gauging 

Mechanical  Powers 
Promiscuous  Questions 


120 
ib. 
121 
122 
123 
124 
ib. 
126 
ib. 
127 


antr 

=  The  sign  of  equality,  and  is  pronounced  equal. 

+  The  sign  of  addition,  pronounced  added  to. 

—  The  sign  of  subtraction,  pronounced  subtracted  by. 

X  The  sign  of  multiplication,  pronounced  multiplied  by* 

-r-  The  sign  of  division,  pronounced  divided  by. 

:  ::  :  The  sign  of  proportion,  pronounced  is  to9  so  is,  to. 

&    Sign  of  difference,  pronounced  the  difference  between. 

v/>  or  5,  The  sign  of  the  square  root. 

«^y,  or  i*  The  sign  of  the  cube  root. 

T2,  Denotes  that  the  number  7  is  to  be  squared. 

83,  Denotes  that  the  number  8  is  to  be  cubed. 

-.-  That  is. 

-  Vinculum,  and  (  )  parenthesis. 


19;  twelve  added  to  7  is  equal  to  nineteen. 
-3=15;  twenty-three  subtracted  by  eight,  equal  fiftee 
9x8=72;  nine  multiplied  into  eight  equal  seventy-tw 
24—3=  3;  twenty-four  divided  by  three  equal  eight. 

Sometimes  the  division  of  one  number  by  another,  is  e 

pressed  in  the  manner  of  a  vulgar  fraction;  thus  %7  = 

uy-seven  divided  by  nine  equal  three.     Multiplicati 

?o  frequently  denoted  by  this  mark  .  thus  7.1W2=:8 

o  twelve  equal  eighty-four 

6  :  9  :  :  8   :   1  2;  as  6  is  to  9  so  is  8  to  1^. 
A  OQ  Z:  .1  -.iifterence  between  A  and  Z. 

v'Si^rO,  or  81i=95  the  square  root  of  81  equals  9. 
10  :  100  •  .•  I   :  10;  as  10  is  to  100  that  is  as  1  is  to  10. 
A  vinculum  is  placed  over  several  quantities  to  deno 
are  to  be  considered  as  one  simple  quantity;  2 
1  for  the  same  purpose, 


JOSEPH  B  HAMMOND 


TO  THE 


WESTERN  CALCULATOR. 


NUMERATION. 

Page  a 
Example  1.    16         2.    49 

Page  9. 
3.  385    4.  2610    5.  64536    6.  253842 

7.  5600006    8.  90000305    9.  829006002 

1.  Five. 

2.  Seventeen. 

3.  Thirty-five. 

4.  Four  hundred  and  fifty-eight. 

5.  Six  thousand,  eight  hundred  and  twenty-nine. 

6.  Seventy -two  thousand,  three  hundred  and  forty-eight. 

7.  Three  hundred  and  eighty-four  thousand,  seven  hundred  and 

twenty-one. 

8.  Two  millions,  six  hundred  and  eighty-three  thousand,  two 

hundred. 

9.  Fifty  millions,  six  hundred  and  seventy-eight  thousand  and 

twenty-four. 


ADDITION. 


Ptgt  10, 


g.  1581366 
4.  2284038452 
6.  2968194198 
8.  5075431465 
10.  3644772*180651 
1* 


3.  2468727092 

5.  2263764650 

7.  5831S33360 

9.  9750563355 

11.  3847099705067 


6 

1    **125 

V*:   /:*?•' 

,      2060 
8009574 
6 

SO  11  804 

5.     297 
125 
754 

127    ' 
245 

si 

JBTRACTtOS, 

Application. 

fage  11. 
"'  .dolls. 

•>.*286 

C    29 
1)672 

barrels    dolls*/ 
4.     60  for     480 
75           675 
220         2200 
126         1386 

10000 
4000000 

4018853 

6.      1209 
476 
8017 
3119221 

S1112 

7.     1811 
21 

481bbls.4741dols 

dolls, 
8.  George      3560 
William   3240 
Samuel     2850 
Henry       2555 
Thomas    2226 

3832 

3128923 

1548 


SUBTRACTION. 

Page  12. 


14431 


3.   44064062638 
5.   56053355680 

1.  1818 
1777 

41 


•Application* 
2.     5648 
3460 

2188 


4.  5390534442 
6,    OOOOOCC1 

barrels 
3.      1260 

sells  to  A     520 
B     435 

sold  in  all    755 
unsold  505 


4.     6000 
600 


cents 

cents 

145 

35  penknife 

75 

25  slate 

40 

64  paper 



30  apples 

given  him     260 



given  away  154 

154 

left 


106 


MULTIPLICATION. 

MULTIPLICATION. 

Case  1.    Page  13. 


2,     24639576 
4 


3.  3675432568 
8 


98558304 
5,  476324753 


20403460544 


Page  14. 

6.  964703024 
6 


7.  74020005 
8 


2334123765 

8.2901946808 
9 

26117521272 


5788218144 


592160040 


9.    246354276 
11 

2709897036 


Case  2. 

2.  Mult.  68523047653 
by       2367 

479661333571 
411138285918 
205569142959 
137046095306 


162194053794651 


Mult,  5221 
by     145 


Page  15. 

4.    Mult.    23430 
bv          230 


26105 

20884 
5221 

757045 

Mult.  89536925 
by  735 

447684625 
263610775 
626758475 


702900 

4085 


9,     3950 

35550 
7900 
$  1145,50 


5388900 

Mult.  78969587 
by  5893 

236897961 
710693883 
6317- 
394829935 

465346561391 
10.    40 

480  pence 


5.  Mult,  3800920 
by    80750 

190046000 
2660644 
30407360 


306924290000 

8.    1.15 
75 

575 

805 

$86.25 


MULTIPLICATION 


18292575 


Case  3.    Page  15. 


2.  Mult.  871075 
by  21    3 

3.  Mult.  2453642 
by  36     6 

4.  Mult.  43102 
by  64    8 

2613225 
7 

14721852 
'  6 

344816 

8 

88331112 


2758528 


5.  Mult.  23645  by  144 
12 


283740 
12 

3404880 


Case  4.    Page  16 


6.  Mult.  12071  by  99 
9 


108639 

11 

1195029 


Mult.  3600 
by   400 

3.  Mult.  44000 
by  550000 

4.  Mult.  663000 
by    60000 

1440000 

220 
220 

39780000006 

24200000000 

NOTE.     Ex.     1.  200X10     =2000 

2.  462X100  =46200 

3.  879X1000=879000 


dolls,  375 


1.  15 

2.  125 

3.  32  cents 

25 

43 

440 

75 

375 

1280 

30 

500 

128 

dolls.    53,75 


dolls.  140,80 


4.     60 X  125=75,00  wheat    5.     100        6.     32 

40x  85=34,00  rye  10      sub.   20X24=480 

S4X  50=17,00  corn  

1000  dimes     12x18=216 

8 126,00  amount  10  

paces  696 

10000  cent* 


DIVISION. 


Page  17. 

7-     7525             8. 

dolls.  250  A's 

g  gio  A 

125 

3 

750  B 

3000  C 

37625 

750  B's 

15050 

4 

4000  altogether 

7525 

.. 

3000  C's 

940625 

DIVISION. 

Case  1. 

3,    4)4756394344 

4.     5)97036142 

5.    8)37846210 

1189098586 

19407228} 

4730776f 

6,     12)64381259 

7.     6)3824966 

8.     7)46825486 

53651044! 


6374941- 


6689355! 


9.     9)8297463813 
921940423! 

Case  2.     Page  13. 

divisor  dividend    quot.        4.    735)47989536925(65291886  quotien 
3.    64)29687624(483869  4410  735 


3889 
3675. 

'l470 

6753 
66*5 

326459435* 
195875759 
457043209 

408 
384 

1855476 
2783214 

247 
192 

556 

512 

29687616 
8  rem. 

29687624  proof 

47989536925  proof 
"Add  in  the  i 

584 

:>76 


8  rem. 


1386     rm'imlo"  as  you  mul- 
735    tiply  when  working 

the  proof. 

6519 

5880 

6392 

5880 

5125 
4410 


715  rem. 


10 


DIVISION. 


Page  18. 


5.    2359)4fi7968967  ( 2084768 
4718  2359 


19996 

18872 


18762917 
10423845 
6254306 


11248  4169537 
9436 


18129 
16513 

16166 
14154 


4917968967  proof 


20127 

18872 

1255  rem. 
7.  175296)19842712000  (  113195 


175296 

231311 
175296 

560152 

525888 

342640 


175296 

679170 
1018763 
226392 
56597ft 
792373 
113195 


175296   19842712000 


1673440 
1577664 

957760 

876480 


6.    671)5374608  (  8009 
5368       ,       671 

~~6608       8018 
6039    56069 

48059 

569 

5374608 


8.    108)5704392(52818 
540  108 


304 
216 


42-2552 

528184 


81280 

Case  4.    Page  19. 

2,     6|00)876154          3.    8jO)2834|7          4. 
146    54rgm.  354    27  rem. 


883    5704392 
864 

199 

108 

912 

864 

48 


16|00)1370|00(85 

128 

90 

80 

1000  rem. 


3.     11000)8761029  4.    110000)8010000 


NOTE. 
1100)256154 

or  256  quol.  54  rem,          or  876  quot.  29  rem.          or  80  quot. 
Application. 

•I.    855)4275(5  boys 
4275 


DIVISION. 


11 


28)2072(74 
196 

112 
112 


Page  20. 

3.  75;45000f60Q 
450 

00 


5.     Here  begin  with  the 
150  and  work  backwards, 
thus  5)150 

50 

12  subtract 

2)18 
9  Ans. 


4.    8164 

10  subtract 

27)8154(302 
81 


54 
54 


6.     13440  dollars 


|=5360  eldest  son's  share 
£-=2688  2d  son's 
£=2240  3d  son's  " 

|=1920  4th  son's         " 
J=1680  5th  son's         " 


11888  sum  of  5  shares 


then  13440 

11888  subtract 


1552  6th  son^s  share 


72084J5190048C72        8.    Because  A  has  10  cents  per  day  more 
504588  than  B,  and  B  10  cents  per  day  more  than 

C,  it  is  evident  that  A  will  get  in  all  8  dol- 
144168  lars  more  than  C,  and  B  4  dollars  more 

144168  than  C,  which  together  is  12  dollars, 

then  from  228 
take     12 

Diyide  by  the  number  of  persons  3)216 

72  C's  share 

consequently  76  B's  share 
and  80  A's  share 

575;i437,50(250 
1150 


9. 


half  Eagle  500 
half  a  dollar  50 
quarter  dollar  25 


575  cents 


2875 
2875 


10.'   Captain   6  shares 
Mate        4      " 
Seamen  40      •" 

No.  of  shares  50 


Then  5|0)455|0 

dolls.    91  one  seaman's  sh, 
Now91x6=546Capt.  share 
And  91x4=364  mate's  share 


FEDERAL    MONEY. 

FEDERAL  MONEY. 

ADDITION.    Page  22. 
2.     Dolls.  19520,43  5.     Dolls.  £04588,00$ 

SUBTRACTION. 
2.    Dolls.  S25216,94|  3.     Dolls.   1267£S.53f 

MULTIPLICATION.    Page  23, 
2.     Dolls.  15823,50  3.     Dolls.  57408,79i 

DIVISION. 

2.     Dolls.  726,55  3.     Dolls.   1034S,79& 

PROMISCUOUS  QUESTIONS. 


E  d  d  cm 
Quest.  I.     2.5,0,0,0.0 

6,2,0,0.0 
8.0.0 
7,5.5 

31,3,5.5,5 
Or,  S  SIS  55J  cts. 

3.  Sugar  g  39,87i 
Coffee  22,181 
Tea  2,m 

S  64,181 
5.     Lent  S  1000,00 


Dolls. 
2.     1055  in  notes 

260       gold 
5650       silver 
250  cents 

S  4~967,50 


4.   .From  S  645,952 
Take      350,00 


„       .      ,    .  f  160,25 
Received  at  » 

sundry  pay -ts. 


Received  in  all  S  746,20| 
Unpaid  g  253.791 


S  295,65? 

6.     S  102,19 
Mult,  by  120 

g  12262,80 

7.     Dolls.  4,50 
16 


3474 
5211 
173766 

$2295,50 


g  72.00 

S.     5)6022,50 
g  1204,50 


COMPOUND    ADDITION.  13 

Page  23. 

10,     18)252.90(14.05  11.     45)£f>.  50(50  cts. 

18  225 

72  0 

.    72 

90 

Page  24. 
12.         25)15535,50(622(1.  22c. 

COMPOUND  ADDITION. 

ENGLISH  SIDNEY.    Page  25. 

&         £136739  6  3£  3.     £15725  11  Hi 

TROY  WEIGHT. 

lb.     oz.  diet.  gr.  lb.     or.,  dirt,   gr, 

$.        22     7     6     6  3.     33     0     9     19 

AVOIRDUPOIS  WEIGHT.     Page  20. 

T.   cwt.qr.ll).    oz.   dr.  -irt.qr.lb.oz.dr. 

1.     128  12  1  18  14  10  2.     2921  3  0  9  0  S 

APOTHECARIES'  WEIGHT. 

tit.    oz.   dr.    sc.    st.  lb.      oz.    dr.   sc.    gr. 

1.         24     4     5     2     16  2.     220     2     0     2     10 

CLOTH  MEASURE. 

Yds.    qr.   na  E.  FL  qr.  na.  E.  Fr.   qr,  rta. 

1.    215     2    0         2.     173     0    0          S.     128     4     0 

E  Ln.  qr.   na. 
4.       221     0     2 

LONG  MEASURE.    Page  27. 

leg*   m-   fur>  P°-  yd-  ft-    2W-    be.  £.  M.  fur.    yds.  ft.  in. 

1.  207  27     5     6     1     0    4     0       2.    13  2     3     167  2  10 

LAND  MEASURE.    Page  28. 

A.   R.  P.  A.     R  P.  A.    R.   P. 

1.     161  1  29  2.     2134  2  23  3.     1999  2  32 

CUBIC,  OR  SOLID  MEASURE. 

,Co.fl.     in.  T.   ft.      in.  T.    ft.     in. 

I,     26  4  1407        2.     21  16  1542        S.     271  26  1294 


14  COMPOUND    BUBTRACTIOX. 

TIME.     Page  29. 

F.  mo.  d.    h.    r.ii.  sec.  Y.     d.    h.  rat  «e«. 

1.     104  £  £6  13  £3  27  2.     14  240  4  4  33 


sis-  de$.  mi.  sec.  sig.      °        '        /> 

1.       9     16    9    17  2.       10     12    45     S3 

LIQUID  MEASURE.    Page  30. 


1. 

T. 

20 

hhd. 
0 

gal. 
15 

qt.     pi 
3        1 

2. 

r. 
79 

hhd. 
0 

S<* 
11 

DRY 

MEASURE. 

1. 

Lu. 
175 

P.. 

#. 

3 

2. 

Lu. 
1553 

P. 
3 

8*. 

4 

3. 

fctt.    p. 

41920  0 

9* 

7 

COMPOUND  SUBTRACTION. 

Page  31. 

T.   c??f .  qr*.  ZZ;.  03.  dr.  mi  /ur  P.  ft.  in.  It, 

2.       156  19  0  22  2  6  5.       124  5  34  4  7  a 

iu.     P.    f  D.       h.     mi.    «cc. 

4      53     3     4     1  5.     83     21     44     S4 

T.  hhd.  galqt.pt.  sig.dcg.  mi.  *cc.  JH.    It.  P 

6.     29  2  .  47  3   t          7.     £  22  15  41         8.     408  2  22 


Promiscuous  Questions  in  Compound  Addition  and 
Subtraction. 


y(h.   qr.  ~na,  £.    s.     d. 

1.     No.  1,     S6     3     2  2.     1st.     6     2    4 

6-     f2,     45"  "l     3  2d.     5  10    9* 

"     3,     48     2     1  3d.     700 

"     4,     52     0     3  4th.    8  10     6 

«     5,     G4    C2    0  5th.    926 

jards  247    21  ^36    6    l|coft 
Received  ^22  10    6 


Note  dSlS  15    7i 


COMPOUND  SUBTRACTION* 


15 


Page  32. 
lb.  oz.dict.  gr. 

Quest.  S.  Bought  26  9  10    0 
Wrought  up  18  0  16  10 

Has  left     ib.  8  8  13  14 


lb.  os.  dr.  sc.  gr. 

4.  Bought  6  10  6  2    0 

Used      4    5  4  I  17 


lb.  2    520    3 


5.     Charles  born         1817 
William  born       1816 

3    20    9  evening 
1     15     6  morning 

Year      1 

Cwt.     qr.       lb. 
6.  1st     18      2       14 
2d.     16      3       18 
Sd.     22       0       24 
4th.    24       1         0 

2      5  15 

7,    foMfo 

c  7.3 

yds.    40   sold  and  left. 

yds. 
Then  from 
talc     40 

Cwt   82      0        0 

Or,  4  Ton,  2  cwt 

yds    loforDandE. 

But  hocnnsc  E  lias  half  ns  muc'h  ns  D,  and  together 
have  15;  therefore  D  has  10,  and  E  has  5, 


8, 


gats. 


bought  1  pipe    =T26 
2  hhds. 

S  quarter  casks 


-1    26 
L  26 


lea 


330    bought 

148 J  sold  and  leaked 


gals. 

sold  1  hhd.     63 
2  qr.  casks  52 
*  fpipe     17 
ed-}  lihd.    11 
I  cask      5| 

gallons  148i 


181  Heft 


9.     Bought    E.  Fr.  qr.  na.  yds.  qr.  no. 


two  first 
two  last 


2=14  1  2 

3     2  =  14  1  2 

2     5=12  2  3 

2     3=12  2  3 


yard*  54    0    2 
40    2    0 


13    a    2  yards  left 


16  COMPOUND    MULTIPLICATION. 

COMPOUND  MULTIPLICATION. 

Case  1.    Page  33. 

T.     cwt.  qr.    ft.     oz.   dr.  76.     oz.    diet,  gr* 

2.     146     -2     3     14     6     1  G.     71     10     17    7 

lit   pc.  ql.  •?.  pi,  deg.  mi.ftar.p. 

4.     199  30        5.   ^3  33  ^  1         6.     67  18  6  52 

SI.     R  P.  bit.  pc.  qt~ 

1  140  8.     809  u  34  9.     533  1  £ 

yrs.    in.  w.d. 
10.      76;-  11.       1508  '2  2  6 


£.     s.     d. 

Qaezt.  1.  2.       1     2    6| 

5  9 


£10     2     SJ 

£.     A        r,7.  JE.     9.      d. 

4.       2     4     2i 
It  12 

^7~0~s]  ££6  10    6 

Case  2.     Page  C4. 
rication. 

T.   cii-t.  qr.     l->.     cz.     c>. 
1.       4     3     1    .16     8     10  by  36 
6 


£5     0     1     15     3  12  product  bj  6 

130    21       76  8  product  by  56 

•£.   .«?.  d  T.  cict.qr.fi>. 

2.    Mult.  1^>  G  9  by  3.    Mult.  21-  4   2   7  bj  43 

G  12 


V/6  29014   3   0  pro.  by  If 

4  4 


0  product  by  24  1162  ID   0  0  pro.  by  4$ 


COMPOUND  MULTIPLICATION.  17 

Case  3.    Page  34, 

d.    h.   mi.  sec. 
Mult.  «    4      12    5  by  29 
*  4 


32  16    48  20  product  by  4 
7 


223  21     S3  20  product  by  28 
8    4     12    5        "  1 


days  257     1     50  25  product  by  29 
Case  4.    Page  35. 

£.  s.  d. 
3.      1     2    3x7 
10 


11     2     oxl 
10 


72   10     0  '      HI     5    Oval,  of  100 

2  1126  :<    <      10 

7  15    9    '    <       7 


145     0     0  value  of  200 
29     0     0     4      of   40  £'130    3    S  val.  of  117 


£174    0     0  value  of  240 

£.    s.    d. 
4.        1     2    6x5 
10 


11     5    0X7 
10 


112  10    0 

2 

225    0    0  value  of  200 

78  15     0     4      of  70 

5  12    G     *      of     5 


£309    7    6  value  of  275 
9* 


13  COMPOUND    DIVISION. 

COMPOUND  DIVISION. 

Case  1.    PageJS. 

£.      8.    d.  '  T.  cwt.gr.  Ib. 

2.     Ans.  187     18    5|  S.     Ans.   15    GOT 

yds.  ft.  in  f  T.  hhd.  gal.  qt. 

4.     Ans.  *192    0    4}  5.     An?  58    2    13    S{ 

w.   d.  h.  mi.   sec. 
6.         Ans.    3    0    16    16    53 

Case  2. 

£.      9.    d.  £.    8.     d. 

2.       11J134    18    8  3.     12J984     0    0 


4)  12     54  quot  bj  11        12)  82     0    0  bj  12 

£3     1    4     «     by  44  ^6  1C    8bjl44 

£.      8.     d. 
4.      12;474     0     0 


6;  39  10     0  quotient  by  1 
M  11     8  •    by  7 

Case  3.     Page  37. 

£.    s.  d.      £.    s.   d.  £.    s.   d. 

2,    34c)iO'^  13   9     (    1      a    9  3.       232     4     » 

--  -,*  -*.   d. 

64  524)4644(8    lCi] 

:>0  4192 

S45)r23P<35.  *      452 

J2 

524)5433(104. 
524 

34D):;105(9J,  193 

3105  4 


248  rem, 


COMPOUND   DIVISION,  ETC.  19 

Page  37o 

&.  9.  d.  £.  8.  d.          £.  i.  d.  £.  $.  d. 

4.  654)323(5  12  4J(  4  18  llf      5.  68)132  0  8(1  18  10 
2616  63 

620  64 

20  20 

054)13412(18*.  68)1280(18*. 

654  6tf 

5872  600 

5232  544 

640  56 

12  12 

654)7684(1  1&  68)680(10<L 

654  €8 

1144  0 

654 

490 
4 


19G2 

Promiscuous  Questions  for  Exercise  in  Compound  Addi- 
tion, Subtraction,  Multiplication  and  Division. 

s.    d  Ib.  oz  dwt.  gr. 

Quest.  1.        2    5X2  2.    4     1     15    22 

_  1£  _  __  11  ingots 

1    4    1X7  45     7"  15      2 

10 


12     1     8 

6 

72  10    0  value  of  600  yards 
89'2*       of  70     * 

4  10     4       of     2     6 

£81     4    0  value  of  6f2  yards 

T.  act  qr  Ih.  T  cict.  qr.  Ib. 

S.    1    2    3    1(5  4.       8;  9    3    0    16 

Mult,  by  8  loads 

0    16 


$0  COMPOUND    DIVISION,    ETC. 

Page  37. 

A.      7?.  P. 

5.     SCO     2     20 

5 


1503    0    £0 
3 

45G9     1     £0 
Page  38. 

A.      K.    P.  bu.    d.    c.        d.    c.    m. 

6,      5)4509    1      20        7.     179;  201  S7i  (  I     12    5 

1 79        or  I  doll.  12|  * 


ryvM 

»>     ^u 

£23 
179 

SCO 

2     20 

447 
S58 

89'5 
895 

• 

8. 

rf. 

7X5 

10 

9. 

Mult,  by 

bushels 
1COO 
10£  cts. 

5 

.10x6 
10 

10000 
500 

£  18     4  SI05,00 


8    15    0  in  200  days 
1    15    0    *     GO     b 
211     *       5     * 


12  11  in  365  daj3 

£     s.    d. 

lit.    cl  c.        (?.  11.    s;47   12  10  J 

155  XS  05  =276  75  sold  for 
135x1  6^=^19  S7.ipnmecost      ^15    17     7i 

"S57  372  gain  ^  1  15    Si 


COMPOUND   DIVISION,  ETC. 
Page  38. 

dolls.  ^  £r- 

19     9708  13.         17    8 

2  9 

5Y19416  716    0  wt.  of  9  dolls. 

a«— — — •—  *^ 

3883  20  eldest  son. 
oz.  39    0    0  wt  of  45  dolls. 

3)5824  80  rem. 

S  1941  60  other  sons,  each 

oz.  dwt.  gr.  wt-  lb- 

4.       84     7    20  i15-         25=2! 

20  Mult,  by       13^113 
dwt.gr.gr.  


I50)l68r    (  11     6  +  3  over 
150         11     6  standard 


187       0    0+8  grains  over 
150 

37 
24 

148 
76 

150)908(6 
900 

8  grains 

16,        35X20  dolls.  notes=700  dolls. 
63    Eagles  =630 

284  dollars  =284 

642  half  dollars  =321 
368  qr.  dollars  =  92 
256xl2|  cents  =  32 

Deposited  Doll?.  2059 

.     Checks  560+820==      1380 

Dolls,        6f9 


COMPOUND    DIVISION,'  ETC. 

Page  38. 

dolls,  eta.     dolls,  cti. 
17.        S6  yds.  at  4    66=167,76  cost 

Add  -29.56  gained 
Must  sell  all  for  Siyr,32 

dolls,  cts.         d  e.      ' 
4  yds.  at    2     33«    9,32 
8yds.  at    5     50=  44,00 

12  yds.  is  sold  for  g53332 

dolls,  efj. 

Now,  From  56  yds.  which  must  sell  for     197   33 
Take    12  yds.  which  brought  53    32 

The  diff.   24  yds.  must  sell  for  g!44    00 

.  doll*. 
But   24)  144  (6  dollars. 

cts. 

18.     12^x5  =  0    62|  beds. 

62A x4=2    50  supper  and* breakfast 

75  for  liquor 
25  X5=l     25  for  hay 

S5         12:}- 

From  6  dollars  take  S5,12£,  and  the  remainder 
is  87s  cts.  =875  mills   , 

But  2:1=25)875(35  qts.=8j  galls. 
And  6  dollars -7-5  travellers  S  1,20=120  cents. 

Page  39. 

/*.  m.  h.   m.  minute* 

19.          12  25xby5^  62     5=3725 
11  SOxby  9  =  103  30=6210 

*  Sum     9935mimites 

Now  T  minutes      * 

Mult,  by    -  75  cents  per  day 

_ 

49675 
69545 

h.   mi. dolls,  cts.  MS. 

8=480  )  745125  (  15     52    S+ 


REDUCTION. 


Page  39. 


gal    qt.  pt. 
£0.   5)1534     1     1 

5)306    3     1  t 


galls.     61     1     1 

PS 


A  1 
B  2 
C  6 


ii.    mi* 
21.    9)114    45 

127*   45rom, 


dolls. 
9)180 

20  A's  share 
20X2=   40  B's 
40X3=120  C's 


DEDUCTION. 
MONEY.    Page  41. 


4)120506  farthings 
12)  30126  \ 
S|0)  251JO  6| 


6. 


10 


^       *.      d. 
480     19    9 
20 

9619 
12 


Page 


5.     10)260  centg 

26  subtract 

234  pence 


7.         12)4658 
2|0)  38|8  2 
^19  8  S 


9)  115437  pence 
^  add 


1282631  cents 

720  cents 
^==72  subtract 

648  pence 


9)648  pence 
72  add 

720  cents 
10.     2)24235  half  pewt 


12)1 211? J 


*4 


REDUCTION. 


Page  42. 


11.      216  French  crowns 
Mult.       99  pence  in  a  crown 

3944 
1944 

12)2 1384  pence 
2]0)  178J2 
89/.  25. 

13.      3752. 
8 

3)3000 
$1000 


12. 


£.      8. 

29    If 
20 

597 
12 


7164 
=    796  add 


7960  cents 
Or,  79  dolls.  60  cents 


1. 

I 


TROY    WEIGHT. 

4)  11 5200  grains 

6)28800 

2|0)  480JO  ( pennyweights 
12/240  ounces 
Ib.  20 


2.        SO  pounds 
12 


3.    «|0)4564|8  dwts 
£282oz. 


f  4)27260  grains 
\  6)  6815 
S|0)11S|5  20 
12)  56  15  20 

4lb.  &oz.  ISdwt.  20$r. 


860  0&. 
20 

7200  dwfr. 

24 

172800  gr. 

Ib.  oz.  dwt.  gr. 
4.         4    3     15     £0 
12 

56  ounces 
20 

1135dwts« 
24 

4540 

2272 

S7260  graini. 


REDUCTION. 

Pas-e  42. 


o. 


dwt.    gr. 
8        6 

24 

198  gr. 
24  spoons 


1. 


396 


4752  grains 


AVOIRDUPOIS  WEIGHT. 


3  tons 
20 

60  cwt. 
4 

240  qr. 

28 


1920 
480 

6720  Ib. 


Tons 
3.        5 
20 

100 
4 

400  gr. 

28 

11200  Ib. 
16 


179200  oz, 
16 


£00  drams 


2.          f  4)2887200  (Jr, 
*« 

(.4)  716SOO 


16 


1 79200  oz. 
(.4)     44800 
23)    11200  Ib. 
4)     400  qr. 
2|0)  10|0  cwt. 
5  tons. 

civt.  qr. 
4.          1     3 

4 

7 
28 

,195  Ibs. 
Mult.  6  barrels 

1176  Ibs. 


2867200  dr. 


REDUCTION. 

Page  42. 

cwt.  or.  Ib.  Ib. 

5.        16    2     14  6.      £8)2876 

—  4)  102  20 

66 

28  25ctttf.2gr.20fl 

532 
133 

1862lb. 

APOTHECARIES'  WEIGHT. 

Ib. 
1.         15 

12 

180oz. 
8 


1440  dr. 
3 

4320  scr. 
Page  43. 

Ib.     .  Ib. 

2.        3  3.         2§  4.     2!0)57600|0grai 

12          <  12 

—  —  3)28800  sc. 

36  oz.- 
8 

283  dr. 


ou 
8 

8)  9600  dr. 

16)240  drams 
15  parcels 

12)1200  oz. 
100  Ib. 

17280  gr. 


REDUCTION. 


£7 


1. 


£50  yards 
4 


CLOTH  MEASURE.    Page  43. 

2.        4)8642  nails 

5)2160  2  nails 


1000  qrs. 
4 

4000  nails. 

524  Ells  Fr. 
6 

4)  1944  qrs. 
486  yards 


260  miles 
8 

2080 
40 

83200 
5J 


416000 
41600 

457600  yds. 
3 


432 Ells  E.Sna. 


16  bales 
S6E.FI. 

96 
48 

576  E.  Fl. 

S 

4)1728  qrs. 
432  yards. 


LONG  MEASURE. 


mi.  fu.  P.  yds,  ft. 
2.   11  7  38  2  2 


£5 
40 

3838 
5* 


1372800  feet 
12 

16473600  inches 


19192 
1919 

21111 
3 

63335 
12 

760020 


2280060  b..c. 


28  UCTION. 

Page  43. 

/,.  fu.  yds.  ft.  i 
C.        3)1267200  feet  4.      3    2    'no    l  i 

L 3  miles 

-:00  yards  g 

8  fur. 
s;  19£0  furlongs  - 

-  74 

GO;  240  miles 

1493 

4  dc^.  149 

.  lo 
3 


49171 
12 


inches 


SCO  degrees  round 
GO  miles 


21600 


8  furlongs 


172300 

£20  yards 


38016000 

Sfect 

11404SOOO 

12  inches 


1368576000 

LAND  3IEASURE. 
25  acres  2.         4|0^  1 7600(0  perch 

4)  4400  roods 

100  

40  1100  acres 


perches 


DEDUCTION. 


Page  43. 
4JOOJ6400|00  perches 

4|0,)  160(0  each  share 
4)  40  roods 
10  acres 


10  acres 
160 

1600  perches 


48000 
400 

48400  yards 
9 

435600  feet 
144 


62726400  inches 
CUBIC,  OR  SOLID  MEASURE. 


1. 


ft.    feet. 
128;  3200  (  25  cords 


Page  44. 


ton.  ft.     feet. 
2.    20X50^=1000 


w.  d.  h.  m. 
8    2    6  20 

7 

58 
24 

238 
116 

1398 
60 


3. 
TOfE. 


ton.  ft.   cu.  in      cu.  in. 
30X40X1728=5=2073600 


da.  h. 

365  6  in  one  year 
24 

1466 
730 

8766 
60 


83900  min. 


Page  45. 


525960 
60 

31557600  sec. 

10  years 

315576000 


yrs.    days        days    hrs. 
3.         1823X3651=665850     18 

w.  da.  hr.  mi.  sec. 

4.        1X7X24X60X60=604800  sec. 
3* 


DECIMAL 


LIQUID  MEASURE,    Page  45. 
4  tuns  pints* 


3  6  hhds. 

8 

pints 


S;4032 
C3j  504  galls. 

8  hhds. 


38x63x8=19152  pints 


DRY  MEASURE. 

t.  2, 


315 

8 

2537 
2 
i>int3 

ADDITION  OF  DECIMALS. 


2)2196 
8) 1093 
4)  137  2 

34  bu.  Ipc.Sqt. 


2. 


Page  47. 
Ans.  2923400,3687073 


5. 


283.604 

4.             ,24& 

490006,003275 

,012 

21,05 

,02 

1,2 

,6 

6200,3476 

,41$ 

,5 

Sum  4965  12,204875 

Sum  1,791 

6. 

125,5 

7.        5,4 

220. 

1000,000005 

15,04 

,003S 

15,072 

100,004 

844, 

2,01 

6000,00004 

2,2 

93680,0004 

300.                      Sura 

10142,582005 

!oo 

Sum  100000,44444 

Sum  1397, 


DECIMAL   FRACTIONS. 


31 


SUBTRACTION  OF  DECIMALS. 


Page  47. 


2.        Ans.  685,49563.2 


3.        Ans.  8,3047 


4.        From  45,005 
Take  23,65482 

5. 

From  629,2 
Take  200.002 

Diff.  21,35018 

Diff.  4#J,198 

Page  48. 

6.    From  5, 
Take     ,10438 

7.    From  2, 
Take     ,00002 

8.    From  16,  ' 
Take      ,016 

4,89562 

1,99998 

15,984 

MULTIPLICATION  OF  DECIMALS. 


Mult.  ,385746 
by  ,00463 

1157238 
2314476 

1542934 

,00178600398 


Mult.  158,694 
by      23.15 
793470 
158694 
476082 
317383 
367376610 


6.    Mult.  ,024653 
,00022 
~49306 
49306 


7.    Mult.    25,04 

,002 


,05008 


,00000542366 


8.    Mult.  645,003 
,000005 


,003225015 


Contraction  in  Multiplication  of  Decimals. 


Page  50. 


3.    23,463  multiplicand 

43,2  multiplier  reversed 


46926 

7039 

938 

54,903 


5.    3,141592  multiplicand 
8347,25    mult.  rev. 

1570796 

62832 

21991 

1257 

94 

25 


4.    234.216  multiplic. 
543,2    mult.  lev. 

46843 

7026 

937 

117 


549,23 


165,6995 


$2  DECIMAL   FRACTIONS. 

DIVISION  OF  DECIMALS. 

Page  50. 

3.   23,7)65531,0(2756,16        4.  64,25)234,70525(3,653 
m  19275 

1792  41955 

1659  38550 


1331  34052 

1J>:>  32125 


1460  19275 

14-22  19275 

380 

5.   3)10, 


1422 
8  rem. 

6.    ,9)9,0  7.     ,00463)5001 78600398(,385746 

1389 

10  

5970 
3704 

2660 
2315 

3453 
3241 


2129 
1852 

2778 
2778 
Page  51, 

8.       2,46),2327898(,09463  9.        ,09463),2327898(2,46 

2214  18926 


984 

m 

1476  56778 

738  56778 

738 


DECIMAL   FRACTIONS,.  S3 

.Page  51. 

10.    ,01S),000162(,009 
162 

Contraction  in  Division  of  Decimals. 

Page  52. 

4,    1,346767)74,33373(55,193  5.    9;3G^4C7}S7,07G326(9,297 

67339 

6994  2767 

£734  .! 

260  914 

135  843 

125  71 

121  65 

4  6 

4 

6.         2,45)32,68744231(13,34 
245 

818 
735 


10 
10 

6,24),0046S72345(.00075 

437 

31 
31 

0 
REDUCTION  OF  DECIMALS. 

Case  1.    Page  53. 
*•     2)1,0(,5  3.    4)3,00(,75 


DECIMAL   FRACTIONS. 

Page  53, 

4.       8)7,000(,875  5.      25)1,00(,04 

6.      60)5r,00(,95  r.       15)6,00(,40  cents 

Case  2, 

s.  £.  d  9 

2.        19-7-20=95  3.         s'-M2=,25 

<y  ^.  cwt.gr.  or. 

3-f-(12x20)=50125      5.    4  S=18--(20x4)=225to» 


gr.   lo  . 


.       .       . 

6.  £    14=70-f-(28x4)=,625cwt. 

^r.  ?2«.  wa. 

7.  3     3=15-5-(4x4)=59375jds. 

Or  thm, 

5.    qr.  A  I  2,         6.    Ib.  28  [  14        7.    net.  4  \  S 
cu;/.  20  I  4,5  4  I    2,5  ^r.  4  |  3,75 

,235  ton.          cwt  ,625  yds.  ,9375 

Case  3.    Page  54. 

^  ft. 

2.       ,75 

20 

shillings  15,00  ounces    8,4 

--  20 

15«.  __ 

dwts.  8,0      8oz.  Sdwts. 

*'     '61Jcwt  5,     ,3375  acres 

_  4 

qr.  2,468  -  • 

rood.  1,3500 

^lb.  1^104  _  ^ 

_  per.  14,0000 

OZ.    j 

1  rood  14  per. 
dr.  10,624 
2qr.131b.lcz.10dr, 


DECIMAL  FRACTIONS. 
Page  54, 


6. 

tun. 
,258 
4 

days. 
7.        ,761 

24 

hhd. 

1,032 
63 

hrs.  18,264 
60 

gal. 

2,016 

mi.  15,840 
60 

Ihhd. 

2  gal. 



sec.  50,400 
18  hr.  15  mi.  50,4  sec. 

Ib.  9.        365,25  days  in  a  year 
8.         ,7  ,3 

12  

—  days  109,575 
oz.  8,4  24 

20  

—  hrs.  13,800 
dwt.  8,0  60 


8  oz.  8  dwt.  min.  48,000 

109  d.  13  h.  48m. 

day   hr.    hours  T.  cwf.  qr.  Ib. 

10.         ,41X24=9,84  11.       ,17  ,19  ,17  ,7 

,16  20 


hrs.  9,68  cwt.  3,59 

60  4 


min.  40,80  2,53 

60  28 

sec.  48,00  15,54 

' 

9  h.  40  m.  48  sec.  Scwt.  2qr.  15,541b. 


36 


DECIMAL    FRACTIONS. 


Promiscuous  Questions  in  Decimal  Fractions 
Page  55. 


2,    ,36  ton 
20 

7^0  cwt. 
4 


28 


quest.  1.    Mult.  ,09 

by  ,009 

Prod.  ,00081 

3,    ,9125  ounces 
20 

dwt.  18;Ix>0 
24 

gr.  6,OOJJ 

18  dwt.  6gr. 


4.     S15)4,00(,0127  nearly 

oz.  diet.  gr. 
5.        2    16    20=1364  grains 

And     1  pound=5760  grains 

Then,  5760)1S64,0(,236S 


23040 
5760 

80ii,40  lb. 
16 


miles 
'  6.      ,1392 

8 

fur.  1,1138 
40 

per.  4,5440 


27200 

2720 

yds.  2,9920 

1  fur.  4  per.  3yds. 

11.     1  doll.=100  cts. 
3 

15)300 
6)  20 

6    cents 


7.  4;3,00f,75 

8.  112;6,00(,  0535714 

9.  365;i09,5(-,3 

10.  ,04X50X1 728 -=5456 


12. 


hhds. 

,875 

63 

2525 
5250 


gall.    55,125 
4 

ql.  ,500 
_2 

pt,  1,000 
55  gal.  1  pint 


SINGLE    RULE    OF   THREE    DIRECT.  37 

Page  55. 

14.    365,25  days  in  a  year  15.     ,73-i-(3x525)  that  is 

>to  ,75),73(,973£ 

18,2625 

24  yr.    w.    d.    h.    mi. 

16.      ,05=2      2     19    12 

730500  j)V 

'J^rOKA  ILI  * 

,5=0      0       0    30 


438,3000  hr. 


60  2t0.2d  ISA.  42m.  * 


26298,0  mi.  A.    R.    P. 

60  18.  ,6X4X40X,  02=1,92 

1577880  sec. 

T.     h.     gal. 
17.       ,  4    ,3     ,8          cub.  in.  ft.       in. 

4  19.    l-;-(i28x  1728)  that  is 

1 ,9  221 184)  l,OOOGGO(,000004-h 

63 

120,5  gall. 

4  20.     (28  deg.  48  mm.)-=-360  deg. 

482,0  qt.  ^at  *s> 

o  mi.         mi. 

%Ipints  21600)1728,00(,08 

SINGLE  RULE  OF  THREE  DIRECT. 

Page  57. 

lb.      d.    C.        lb.     T 
Quest.  2.  As  112  :  12,32  ::  16  :  - 


/6.     cts.       lb.     cwt. 
3.     As  1   :  36  ::  336  (=3)  :  120  dolls.  96  cts. 

yds.  yds.  yds.  yds.  yds. 
4.        23 -f  24  +  25+27=99  then 

yd.    cts.     yds. 
As  1  :  72  ::  '99  :  99X72=71  dolls.  28  cts. 

*  This  answer  is  obtained  by  reckoning  12  months  to  the  year,  4 
weeks  to  the  moath,  7  days  to  a  week,  &c.  But  at  52  weeks  to  the 
year,  the  answer  will  be  2w.  4d.  4h.  18rai.  And  at  365£  days,  it  will 
be  2w.  4d.  5h.  48mi.  the  true  Answer. 


S3  SINGLE    RULE    OF   THREE    DIRECT. 

Page  57. 

Ib.  cts.     Ib. 
5.  As  4  :  48  ::  512  (4  cwt.  2  qr.  8  Ib.)  :  61  dolls.  44  cts. 

Ib.    cts.      Ib. 
6.     As  1  :  8  :  :  128  :  10  dolls.  24  cts. 

pair  d.  c.      pair 

7.  As  114  (=9i  doz.)  :  68,40  ::  3    :  1  doll.  80  cts. 

bu.      d.  c.      bn. 
8.    As  20  :  9,60  :;  3  :  1  doll.  44  cts. 

cts.    yd.      d.  cts. 
9.        As  75  :  1  ::  16,50  ;  22  yards. 

c.  qr.  Ib.       d.  cts.       oz.  cts. 
10.  As  32080  (=17  3  17)  :  320,80  ::  6  :  6 

Page  58. 

oz.         Ib.      dol.         oz.     QTv/t  5     d.    c. 
11.        As  116,4  (=9,7)  :  97   ::  1,5  :      ~=l   25 


acres     dolls.  acres 

12.     As  125,5  :  627,5  •.•  1  :  5  ::  4,75  :  23  doll.  75  cents. 

gal.  dolls,  cts.      gal. 

13.  As  1,5  :  4     50  ::  378  (=1,5  tuns)  :  1134  dolls. 

d.  cts.  d.  cts.  d.  cts.  d.  cts. 

14.  First  1  66  +  1  97-f2  31=5  94  the  price  of  I  ream 
of  each  sort.     Then  say  — 

d.  CtS.  of  each  sort.  rf.     cts. 

As  5  94   ;    1    ::  528  66  :  89  reams  of  each  sort. 

Ib.       T.        d.          Ib.     qr.  Ib. 
15.     As  2240  (=1)  :  224  :;  42  (=1   14)  :  4  dolls  20  cts. 

d.  c.    fibl.        d.    c. 

16.  As  5  50  :  1  :;  1402,50  :  255  barrels 

da.         d.  cts.      da. 

17.  As  365  :   1186,25  ::  1  :  3  dolls.  25  cents. 


SINGLE    RULE    OF  THREE   DIRECT. 

Pag!  58. 
da.     d.  cts.         da. 

18.     As  1  :  2  25  ::  365  :  821  dolls.  25  cents,  the 
§um  he  spends  in  a  year. 
Now,  821  dol.  25  cts.+S78  dol.  75  cts.=1200  dolls. 

T.   cwt.  qr.    Ib.          Ib. 

19.     4     10  1     12=10120 

Ib.  cts.            Ib. 

Then,  as  112  :  1,12  ::  10120  :  101  dolls.  20  cents 

20.     4ft.  6  in.  =  54  inches 
$  of  54=27  add 

81 

27 

2187 


19683  solid  inches 

eu.  in.       cts.          cu.  in. 
Then,  as  1728  :  110  ::  19683  :  12  dolls.  53  cts.  nearly 

in.      in.       in.      in, 

21.     (28+14)  X14X3,5=205S  cubic  inches 
cu.  in.        cts.         cu.  in.      d.  cts.  m. 
Then,  as  1728  :  190  ::  2058  :  2  26  2>+ 

£.    s.       s.  T.  cwt.  qr.  Ib.         Ib. 

22.  Oneton=2240lb.  22  8=448  &  203  9   3   3=455815 

Ib.        s.  Ib.  s. 

Now,  as  2240  :  448  v  5  :  1  :  :  455815  :  9116S-=£4558  3s. 

d.   cts.  yds.  d.     cts.      yds. 

23.  As  11  25  :  5  that  is,  as  225  :  1  :  :  850  50  :  378  in  all 

And,  as  18  pieces  :  378yds.  ::  1  piece  :  21  yards 

hf.  yds.    yds.          d.  £.   s.    d.      hf.  yds.  d.     s. 

24.  As    25(=12i)  :  450  (=  1  17  6)  ::  2  :  36=3 

ft.  ft.      ft.  in. 

25.  As  7  :  4  ::  218  9  :  125  feet 


40  SINGLE    RULE    OF   TRHEE    DIRECT, 

A.  R.P. 

£6.    476  5  28=76308  perches.     Then  say 

P.  d.       c.          P. 

As  76308  :  4292  S2i  ::  160  :  9  dollars 

da.     cts.        da.        d.    cts. 
£7.    As  1  :  214  :;  S65  :  781  10  spends 

dolls,  cts. 

Then,  from     1333  00  annual  income 
take       781  10  yearly  expense 

S551  90  he  will  save 
Page  59. 

bu.        d.    c.       bu. 

£8.     As  321  :  240,75  ::  1  .-  75  cents 

na.     yds.     cts.      na.  qr.  na. 
29.     As  24(=U)  :  250  ::  6(==1     2)  ;  62|  cents 

gal.      gal.    gal. 
50.     120i  +  124+126|=37U  gallons 

gal.    d.     s.  d.        gal.        pence       £.    s.   d. 
As  1  :  66(=5  6)  ::  371$  :  24502^=102  1  10i 

7ui.  da.  mi. 

31.  12  x  5  =60  the  distance  that  A  has  gone  before  B  starts 

mi.   mi. 

16  —  12=4  miles  B  gains  on  A  per  day 

mi.  da.     mi. 
Then,  as  4  :  1  :;  60  ;  15  days 

£,.      d.        s.  d.          £.        pence       £. 

32.  As  1  :  150(=12  6)  ;;  1000  :  150000=625 

men   bbls.      men      bbls.         bbls. 
S3.     As  365  :  75  ::  500  : 


54.     This  is.  properly  a  question  belonging  to  the  rule  of 
Three  Inverse,  stated  thus, 

cts.       3.         cts. 
As  575  :  360  ::  250  then, 

S75X5GO     ff/f. 
--  =540  acres, 
250  cts. 


SIKGLE    RULE   OF  THREE   INVERSE.'  41 

Page  59. 

mi.      hrs.        mi.        deg.    mi.       mi.  mi.  fur. 
S5,     As  1440(=24)  :  25020(=360x69i)  ::  1  :  17    3 

SINGLE  RULE  OF  THREE  INVERSE, 

Page  60. 

men    days    men  men  days  men 

Quest.  2,     As  60  :  100  ::  20         3.    As  65  :  4  ::  5 
60  4  < 

2lO)GOO|0  5)260 

300  days  52  days 

per.     days        per.  fiy04 

4.  As  6  :  24  ::  9  (=6+3)  :  :i— =16  days 

cwt.       miles         cwt.     1  y  1  *)() 

5.  As  1  :  150  ::  6  :  -—— =25  miles 

o 

ft.  wide        yd?,  long  ft.  wide         QA  V  ^Ofl 

6.  As  80     :     300     ::     60     :    : 

r.  1.        r.  w.          r.l        fiOv^n       fiv^ft 

7.  As  80  :  30  ::  70  :  —?     ^p~34r.  4ft.  8|m. 

foot       feet  feet        1 0  f}(\ 

8.  As  1  :  12  ::  ,75  :~   ~  =16  feet 

37o 

_ydi.  yds.  yds.  T1v40  1 

9.  As  ,75  :  42,5  ::  1,25  :  'I^pff.,25,5  yards 

«^«0 

men     months        men  Iflvd  ^ 

10.  As  10  :  4,5  ::  15  (=10+5)  :  '  .=3  months 

dolla.       yrs.          dolls.        fin  v  1  <? 

11.  As  80  :  15  ::  600  :  ^^=,2  years. 

600 

hours     dap       hrs.         10y4 

12.  As  12  :  4  ::  16  :  il±Z=3  dajs 

days         nun  days      ^Ovdfin 

13.  As  30  :  400  ::  50  :  ^±^=240  the  number  of 

50 

men  the  provisions  will  serve  fifty  days 
men    men 

And  400—240=160  men  must  depart' 
4« 


SINGLE    RUX^E    OF   THREE    INVERSE 
Page  61. 

dolls.       mtf.         dolls. 

14,     As  292  :  6  ::  806  : 

men       mo.         men 

15,     As  1200  :  9  :  i  1600(= 1200 +400)  :  • 

IbOO 

•tin, 

men  cz.  men          t  pnf)  y  1 4       oz. 

As  1200  :  14  ::  1600  :  —-;-—  =  10i  allowance  per  day 


Aad  I4oz.  —  lOloz.  =5%oz.  diminution  required, 

rod.        rod.        rod.        4ft  y  4 

16.     As  40  :  4  ::  25  :  -l-L^    rods. 


in.          in. 


17.    As  12  :  12  ::  3  :  —  —-^48  inches. 


cws.      days         cws. 

18.  As  6  :  91  ::  21   :        ±=26  days. 

^t  i 

mi.  hrs.         pipe        min.        fiOO 

19.  As  600(=10)  :  1  ::  24  :  -^=25  pipes 

inches         ft.         yds.         ft  in.         yd.      O 

20.  As  216(=18)  :  10(=30)  :  :  l,8(=i)  :  ^ 


Q 
18 

in         in.  It.  ra.          TTyOnft 

21,     As  ,75(=|)  :  208  ::  39  :  '^^=4/6, 

3«7 

da.  ml.    mi. 
£2.     5  x20=  100  A  has  gone  before  B  starts. 

mi.   mi. 

25  —  20=5  miles  B  gains  on  A  each  day. 

mi.    da.     mi.      da. 
Now,  as  o   :  1  :  :  100  :  20  B  will  overtake  A. 

da.  mi.    mi. 
And  20X25=500  the  distance  B  must  traveL 

GENERAL  RULE. 

Page  62. 

mo.      mo.         men       800X2        men 

Quest.  S.   As  5  :  2  ::  800  :  -—--=320  the  number  of 

9 

men  the  provisions  will  serve  for  five  months. 
Then,  800  men  —  320  men=?4SO  men  must  depart. 


SINGLE   RULE   OF  THREE. 

Page  62. 

galls,  galls,  galls. 
4.  63 — 9=54  remains 

dolls,  dolls,  dolls. 

And  1194-12=131  must  sell  for 

gal.    gal.       dot.  d.    c.    m. 

Then,  as  54  :  1  ::  131  :  V*=2  42  5|£ 

mi.       mi.  lb.          lb. 

5.     As  64  :  512 that  is,  as  1  :  8  ::  225  :  1800 

dolls,     dolls.  cts.        cts* 

6-     As  1750  :  10  v  175  :  1  ::  8750  :  50 

Promiscuous  Questions  in  Direct  and  Inverse  Proportion. 

zmi.    mi.    hr.      %mi.        mi.      hr.  mi. 
Quest.   1.  As5(=2^)  :  1  ::  246  (=123)  :  49  12  going 

5 mi.    mi.    hr.      kmi.    hrs.  mi.  sec. 
And,  as  7  (=31)  :  1  ::  246  :  35    8    34f  returning 

hrs.   mi.     sec. 
But   49      12        0 
35        8       34f 

Sum  84/i.   20mi,  34f  sec. 

Page  63. 
dolls.        da.         dolls. 

Quest.  &     Stated  thus,  as  1000  :  189  ::  650  Inverse  pro. 

dolls.        dolls*        days. 
Or,  as  650  :  1000  ::  189  by  the  general  rule 


1000X189     20X189 


cwt.   qi\    Ih.  casks  lb.    cwt.    lb.        cwt.  lb. 

' 


ibs.        ct».          ibs  12,60X2016 

Then,  as  112  :  1260  ::  2016  :    -^-j^—  =226d.  80c. 

Ibs.  ct«.  lb.        1060      45 

And,  as  112  :  1260  ::  1   :  -----=—=11  cts.  2|  ms, 

1  1/6          T: 


SINGLE   RULE    OF   THREEo 

Page  63. 


cwt.  qr.  Ib.  chests  Ib.  chest.  Ib 
4.        (1     0     1  4)  X4=126x4=504  the  whole  weight. 


dolls. 

cts. 

Now, 

as  I 

Ib. 

:     80 

cts.  :: 

126 

Ib. 

:  100 

80 

1 

Ib. 

:     90 

cts.  :: 

126 

Ib. 

:  113 

40 

1 

Ib. 

:  105 

cts.  :: 

126 

Ib. 

:  132 

30 

1 

Ib. 

:  125 

cts.  :: 

126 

Ib. 

:  157 

50 

8504    00  the  amt 
Then,  as  504  Ib.  :  504  dolls.  : :  1  Ib.  :  1  dollar 

5.  5oz.x  12= 60  ounces  of  bread  in  a  dozen  of  rolls* 

And,  as  5  :  4  : :  60oz.  :  4Soz.  the  weight  of  flour  in  a 
dozen  of  rolls. 

oz.      cwt.    dolls.  oz. 

Then,  as  1792  (=1)  :  £24  •.•  8  :  1  ::  48  :  6  eta. 

dolls,  cts. 

6.  Cost                 780  00 
Freight               37  70 
Other  charges    30  60 
Gain                 143  00 


8991     SO  cts.  must  sell  the  whole  for 

bbl        dolls,     cts.        III. 
Then,  as  270  :  991     30  ::  1  :  3  dolls.  67/7cts. 

7.        Half  a  ton=10  cwt.  =40  qrs. 

horsei      qr§.        horses    4QX7       7 

As    80  :  40  ::  7  :  ~3[J-=2=3(lr8-   14  Ib. 

Quest.  8.  12ft.=144  inches,  and  9ft.  3in.«=lll  inches, 
144in.xlll  in.  =15984  inches.  In  that  distance  the  large 
wheel  will  have  made  111  revolutions  and  the  smaller  144 
turns.  But  144 — 11 1=33  turns  of  the  less  more  than  the 
greater  in  that  distance.  Now,  as  33  turns  :  15984  inches 
distance  ::  1000  turns  :  484363-^  inches;  this,  when  re- 
duced to  miles,  is  7  miles,  5  furlongs,  34  yards,  1  foot, 
7^  inches. 


SINGLE    RULE    OF   THREE.  45 

Page  63. 
Quest.  9.         This  question  being  inverse,  say 

hrs.         da.          hrs.         15X18        15X3 

As  15  :  18  ::  12  :  — — - = =22£  days. 

.  \.AI  z 

yds.    dolls,  cfs.  yds. 

10.  As  42,5  :  191,25  •.•  1  :  4,5  ::*15  :  67  dolls.  50 
cts.  what  15  yards  cost;  but  f  of  67  dolls.  50  cts.=45  dolls, 
the  amount  that  15  yards  sells  for. 

Now,  4^2  yards  at  1  dollar,  comes  to  42  dollars  50  cents, 
whole  gain. 

d.      cts.      d.  cts.      d.    cts. 

191  25+42,50=233,75  must  get  in  all 

Subtract  45,00  received  for  15  yards. 

Diff.  8188,75  the  sum  that  the  remaining 
27!  yards  must  bring.     But 

yds.       d.    c.       yd.    d.  e. 

As  27,5  :  188,75  ::  1  :  6,86T\ 

11.  State  by  the  general  rule 

yds.    yds.    ft.  ft. 

As  60  :  10  (=30)   v  6  :  1  ::  18  :  3  feet 

rod.     rod.  rod.    A.        rod.    rods. 

12.  As  40  :  640  (=160X4)  ::   1   :  16 

rod.      rod.  rod.    rods. 

Or,  As  40  :  160  v  1   :  4  ::  4  :  16 

Page  64. 

13.  Because  the  large  wheel  has  70  cogs,  and  the  small 
one  52,  the  small  wheel  will  make  70  revolutions  while  the 
large  ones  makes  52.     But  70 — 52=18  revolutions  that  the 
small  wheel  will  gain  in  the  same  time.     But 

rev.        rev.          rev.          £Q  y  i  no      revolutions. 

As   18  :  52  ::  100  :   -     — =288f 

1 8 
feet.      sec.       feet 

14.  1142X60=68520  the  dist.  sound  goes  in  a  minute. 

pul         feet  pul.         feet         m.  f.  yds.  ft. 

Then,  as  70  :  68520  ::  20  :  I9o77|=3  5  145  2| 
dolls. 

15.  5+4+3  +  2+ 1=  15  the  cost  of  1  yard  of  each  sort 

dolls.  dolls,  cts.     yds. 

Then,  as  15  :  1  ::  532,50  :  35£ 


As 
As 

in. 

,5 

Ib. 
90 

in. 

:  36   : 

Ib. 
:  5  v 

ib. 
:  5  : 

18  : 

3 

6x5 

:3 

60 
2 

pounds, 
indies 

1 

in. 
::  36 

6  SINGLE   RULE    OF  THREE. 

Page  64. 
16.     Stated  by  the  general  rule  it  will  be 

mo.    mo.     dolls,    dolls. 
As     1  :  12  ::  127  :  1524 

17. 

18. 

19.     y=,8  and  2 =,5  therefore  it  will  be 

yds.      dolls.        yds.       d.    c.    m. 
As  1   :  2,7118  ::  67,5  :  183  04  6£ 

day    s.  d.  qr.          days. 
20.        As    1  :  16  5  1A5T  ::  365 
12 

197 
4 

789 
365 

3950 
4735 
2367 


4)288000* 

*Here  omit  multiplying  by  the  third 
[12)    72000        number,  because  you  would  immediate- 
-        ly  have  to  divide  by  the  same  number, 
2|0)  600|0        to  bring  farthings. 

300*. 

yds.qrs.ft.  ft.      sq.ft. 
£1.        11x3=33x21=74,25  in  a  piece. 

ft.    ft.     ft.     ft.    ft.     sq.feeL 
And  2  (25  + 


sq.ft.       040       sq.ft.      sq.ft.     ?  q.  ft 

But    840  --  =  340—84=756  in  the  walls. 

sq.ft.   piece  sq.  ft. 
Then,  as  74,25  :  1  ::  756  :  lO-ft  pieces 

ft.     ft.  in.          ft.  in. 
22.     As  50  :  50  10  j  ::  1287  4,  that  is 

in.       in.  in.          in.  ft.     in. 

As  600  :  610^  ::  15448  :  15718fJ=1309  10}J 


DOUBLE   RULE   OF   THREE.  47 

Page  64. 

h.pt.     gal.  h.pt.      gal. 
23.    As  16  (=1).  :  15  ::  100  :  93|  gallons 

DOUBLE  RULE  OF  THREE. 

Believing  that  it  will  be  acceptable  to  some,  I  here  in- 
sert two  rules  for  stating  the  Double  Rule  of  Three,  in  ad- 
dition to  that  in  the  W.  Calculator. 

RULE  FIRST.  Place  the  three  conditional  terms  in  the 
following  order:  that  which  is  the  principal  cause  of  gain, 
loss  or  action,  possesses  the  first  place;  that  which  denotes 
space  of  time,  or  distance  of  place,  the  second;  and  that 
which  is  the  .gain,  loss,  or  action  the  third;  then  place  the 
other  two  terms,  which  move  the  question,  under  those  of 
the  same  name,  and  if  the  blank  space  falls  under  the  third, 
multiply  the  three  last  terms  for  a  dividend,  and  the  two 
first  for  a  divisor:  but  if  the  blank  fall  under  the  first  or 
second  place,  multiply  the  first,  second,  and  last  terms  to- 
gether fora  dividend,  and  the  other  twcr  for  a  divisor;  and 
the  quotient  will  be  the  answer. 

RULE  SECOND.     Work  by  two  statements  of  the  Single 
Rule  of  Three. 

Direct  Proportion. 

Page  66. 
Quest.  2.         Horses  18 


Horses  18  >lobughC  60  horses 
days     203  £  36  days 


Then  _ioX3x2==60  bushels. 


Or  thus, 
hories     days          bu. 

As  18  :  20   ::  10     Then  ;=  60  bushels. 

60  :  36  18*20 

Or,  by  two  statings, 

horses  bu.    horses 
As  18  :   10  ::  60  :  S3|  bushels. 

days     bu.      days  bushels 
And,  as  20  :  33]  ::  36  :  60  as  before. 


48  DOUBLE  RULE  OF  THREE. 

Page  66. 
Stated  thus:  Or  thus, 

3.    Men    7?  ^  1K    521  mei1  men  daw  Ibs, 

days  14  5  °       ''  £   Sdays  7—14—56 

21—  S 

Then  —=4x3x3=36  pounds 

/  x  J.  * 

Or  by  two  stating*, 
men    men        Ib.          Ib. 
As        7  :  21  ::  56  :  168,  and 

days    days        Ib. 
As         14  :  3  ::  168  :  36  pounds. 

Stated  thus,  Or  thus, 

4.     Students  81  dolls.  f!2  stud.  stu.  mo.  dolls, 

I  S84  1  8—  6—384 


i 


months  6  J  (,10  mo.  12—10 

Then,  384g1xS6Xl°=48x2xlO=960  dollars. 

Or  by  two  statin  gs, 

stu.     stu.      dolls.      dolls. 
As     8  :  12  ::  384  :  576,  and 

mo.    mo.       dolls. 
As    6  :  10  ::  576  :  960  dollars. 

Stated  thus,  Or  thus, 

5.    Cwt.  201  dolls.  f40  cwt.  cwt.  mi.  dolls. 

I    25    ]  20—  50—25 

miles  50  J  (.lOOmiles  40—100 

Then,25X40XlQ°-25X£X2=100  dollars. 
20X30 

Or  by  two  statin  gs, 

cwt.      cut.     dolls,    dolls. 
As  20  :  40  :  :  25  :  50,  and 

miles    milds      dolls. 

As  50  :  100  :  :  50  :   100  dollars. 
Stated  thus,  Or  thus, 

6.    Dolls.  7001  dolls,  f  400  dolls.          dolls.,  mo.  dolls. 

L     14   \  700—  6—14 

months    6J  ^60  mo.  400—60 


The.,  =2x4xlO=80  dollars. 


DOUBLE    RULE    OF   THREE.  49 

Page  66. 

Or  by  two  stntlngs, 

dolls,     dolts.      dulls,  dolls.  mo.     wo.    dolls. 

As  700  :  400  ::  14  :  3  and,  As  6  :  60  ::  8  :  80  d. 


Statement,  Or  thus. 

7.    Men  41ro(lsfS  men  men  days  rods 

I  12  <  4—  6—12 

days  6  J          (.24  days  8—  24 


Then,     ^  -=2x8x6=96  rods. 

4x6 

Or  by  two  stalings, 

men  men     rods     rods  da.    days      rods 

As  4  :  8  ::  12  :  24  and,    As  6  :  24  ::  24  :  96  rods. 


Inverse  Proportion. 
}")  days  f  1 

rl 


Stated  thus, 
Quest.  2.    Men  4  inverse")  days  f  16  men  inverse 


Dolls.  24       J          139*  dollars. 

Or  thus, 

men    days    dolls. 

4 3 24 

16 384 

Then,  5X4x384=3x4xl  =  1 
16x24 

Or  by  two  statings, 

dolls,    dolls,     days  days  men    men     days    days 

As  24  :  384  ::  3  :  48  and,      As  16  :  4  ::  48  :  12 

&    Dolls.  241  men  f  93  dollars  men  days  dolls. 

y   4    <  4—31.04 

day  3  inverse  J          (/inverse  16  days.          16 — 96 

^.    •     4X96X3     4X4X3 

Then, =  -       -=3  men. 

24X16          16 

Or  by  two  stating?, 

dotts.    dolls,    men   men  days  days     mm    *nen 

As  24  :  96  ::  4  :  16  and,        As  16  :  3  ::  16  :  3 

5 


50  DOUBLE  RULE  OF  THREE. 


Page  67. 

Stated  thus,  Or  thus, 

4.    Acr.  841  men  f  100  acres  men  days  acra 

[    7   <  7—12—  84 

inv.  days  12  J         (^5  days  inverse  5  —  100 


84X5  1X1 

Or  by  two  statings, 

acres       acres      men    men  days    day$      men    mm 

As     84  :     100  ::  7  :  8£  and,     As  o  :  12  ::  81  :  20 

Stated  thus,  Or  thus, 

5.     Men  inv.  71  days  f  20  men  inv,  men  days  atrn 

\   te  \  7—12—  84 

acres    84  J           (^  100  acres  £0  —  0  —  100 

12X7X100     1X1X5 
Then,  -  =  -  =:>  days 
20x84  1X1 

Or  by  two  statings, 

«crcs     days       acres      days  men  men     day* 

As  84  :  12  ::  100  :  14f  and,    As  20  :  7  ::  14f  :  5  daji 

Stated  thus, 
6.     Inverse  200  lb.~)  miles"!  20200  Ib.  inverse. 

I   40     L 
40  cts.  J          J  6060  cents. 

Ibs.      miles       cts. 

'.-*-.    Or  thus,     200—40—    40 
20200—  0—6060 
mi.      Ib.        cts. 

^        40X200X6060     CA     ., 
Then,  -  -  =60  miles. 
20201b.x40c. 

Or  by  two  statings, 

els.     mi.       cts.       miles 
As    40  :  40  :  :  6060  :  6060,  and  . 

Ib.          Ib.       miles 
As    20203  :  200  ::  C060  :  60  miles. 
Stated  thus,  Or  thus, 

7.    82001  w.da.  f  500  dolls.  men  w.    d.   dott*. 

1  22    6<{  5—22     6—200 

inv.  men  5  J  /  12  men  inv,          12  —  0  -  300 


Then,  &   6rfq.)x300X5jl4  weeks  2  d 
200X12 


DOUBLE    RULE   OF   THREE. 

Page  67. 

Or  by  two  statings, 
dolls,  dolls,     w.  d.     w.  d. 
As    200  :  SOO  : :  22  6  :  34  2  and, 
men     men        w.    d.        w.     d. 
As     12     :    5     ::     34     2     :     14    2 

Promiscuous  Questions. 
.        Stated  thus,  Or  thus, 

12  oxen>  acres  C24  oxen    o^e;z  daysacret 
8  days  5    10    £48  days       12—  8—10 


24—48—  0 


_,        10x24x48     1on 
Then,— -^-=120  acres. 


Or  by  two  statings, 

0#.     oa?.    acres  acres          days  days  acres 
As  12  :  24  ::  10  :  20  and,  As  8  :  48  ::  20  :  120  acres 

wt.       wt.       ivt. 
«.     8000 — 4500=5500  and  9  days — 6  days=S  days 

4500  cwt.>  horses  {  3500  cwt. 
inrcrse  6  days  3     18     (3  days  inverse 
horses    days      cwt. 

Or  thus,     18— -6 4500 

0 3 35CO 

_,         18x3500x6          . 

Then,  =28  horses. 

4500X3 

Or  by  two  statings, 

twt.      cwt.      hor.    hor.  days  days  hor.  hor. 

As  4500  :  3500  ::  18  :  14  and,      As  3  :  6*::  14  :  28 

cwt.  hhd.  cwt.        cwt.   bbl.    cwt. 
3.     12x9=108,  and  2,5x50=125. 

Stated  thus,  Cwt.      toi.     dolls. 

108  cwt.  £  dolls.  C  125  cwt.          Or  thus,  108-460—100 
€0railes5    100  ^300  miles  125-300 —    0 

Then   100xl£5x50°     •  «      


108x60 

>  o  uuiia*    i 

i  UT^Y  uems 

Or  by  two  statings, 

mil.        mil. 

dolls. 

dolls. 

As    60     :     300 

::     100 

:    500,  and 

cwt.         cwt. 

dolls. 

dolls,    d*. 

As     108     :     125 

:  :    500 

:    578    70-Jf 

5£  DOUBLE    RULE    OF   THREB« 

Page  67. 


yds.  $r 
S— o- 
5  qrs.  S    1   ?  4    qrs.  45—4—0 


4.      3  yds.  lib.  545yds.  O, 

s.  S    1   " 


3x5 

•  By  two  statings, 

«      yds.    yds.      Ib.       Ib.  qrs.   qrs.       Ib.        #. 

As     3  :  45  ::  1   :  lj  and,        As  5  :  4  ::  15  :  12 

Or  thus, 

5.     240  miles  7  days  C720  miles          mil.  dayshr*. 
inverse  12  hours  <J    1.2    £l6hrs.  inv.        240 — 12 — 12 

7,20—  0—16 

12X720X12     12X3X12 
Then,  -  — = —        -  =27  days. 

240X16  16  J 

By  tu'o  statings, 

miles     miles      days    dmjs  hrs.     hrs.     days 

As  240  :  720  ::  12  :  36  and,  As  16  :  12  ::  St>  :  27  dajfc 


6.  16,5  feet!  doll.   cts.  fSO  feet 

1,5  feet  L      1      25  <  ^  feet 
1     footj  (_  4.5  feet 

1.25X30X26X4.5    d  !K    c's-  . 
The"'       I6,oxl,oxi      =1-^Tr 

7.  S  feetl  f  SCO  feet  lor^ 
4  feet  ll  corcN  10    feet  high 
4  feetj  [36    feet  broad 

Then,  ^OOX10X36=5G£,  cord3 

,         ?X4X4 

8.  10 :  yards  ^lb.  ^  100  yards 

6  qrs.     3    3    ^      3  qrs. 

,_.          3X100X3      _ 
rhen^  — ——  —  =15  pound. 


Page  63. 

9-     Inv.  24  men       ^  f  6  re  en  iir 

!'>ns;    (dnysj  -°  ;'' 
liigh    f  SU  ^     6  feet  i 
G  ft.  thick  J  L  4  ^e'  thick 

iien,  80x24x20x6x4_ 


4  PUACT1CE.  5$. 


\  .  Pago  G8. 


\ 

9  persons)  doll*.  C. \4  (=9+5)  persons 
5  months  5    450   (  8  months 

450X14X8 

Then  - 


11.       8  persons!  doUx.  f!2  (=S-f4)  persons 

1  month    >  11]  <!    6  months 
10  dollars  J  |Jl  dollars 

114x12x6x11      dyh-  ctV 


12.     22,5  feet  long  T  f45  feet  lon£ 

17,3  feet  wide   |  day3       34.6  feet  wide 

10,25  ft.  dee;>    ^  2,5    <J  12,3  feet  deep 
Inverse    6  men  I    9  men  inverse 

Inverse  12,3  hours      j  (^8. 2  days  inverse 


<ow/N-T.,X34.GX12,3x6xl-2,3 , 
Then  -—— — . — -.  - — — -  by  cancellinp:* 

O  J)  *»  \x  IT  i-  \/  :  n  -?  -i  \x  O  \x  y  O         •»  o 


10,25  10,^5 

PRACTICE. 

Case  1.     Page  GO. 

1,75  c/*.  |      |    250 

250  yards  Or, 

,50 


8750 
350  8437,50 


S437,50 

•  Cancelling.    That  is,  when  tlio  divisor  and  dividend  are  the  cott 
tinued  product  of  a  number  of  factors,  you  can  divide  both  by  thesan 
divisor,  and  the  operation  will  !>»•  fjr^atly  Hbridjjed. 

Thus,  in  the  above  example.  T  oh-erve  tlmt  4^,5  goes  into  45  twice, 
without  a  remainder;  that  17,  :i  is  oo>ita!n:jd  twice,  i»  34,6;  and  that 
SX8,2  goes  once  into  6X  1*2,3.  Now,  by  dividing  the  numerator  aud 

" 


.     .; 
denominator  by  these  divisors,  we  obtain  --  ...  ,;r  ---  as  above 


54 


3.     201  yds, 
4,20          ^ 

4020 
804 

8844,20 

4.     2210  yards 
X  1,10  cents 

82431,00 

5.  2,415 
X  421  yards, 

2415 

4830 
9660 

1016,715 


FRACTION 
Pago  6& 


Or,    201  yds. 
4 

804 
cts.  is|=  40,2 


cts. 
Or,     10 


Or, 


cts. 
20 
20 

10  ms. 
5 


Or,  101 6  dolls,  nets.  5ms. 


6.     625 
25 

3125 
1250 

8156^5 

>t 
7.     8275 


8844,20 

2210  yards 

221 

2431  dollars 

421 
2 

842 
84,2 
84,2 
4,21 
2,105 

1016,715 
Or  thus, 


cts. 

25  is  |    625 


8156,25 


ms» 


33100 
33100 

36410,0 


8.     8275 
5 


Or,  364  ddls,  10  cts. 


41375  mills. 


A     ,., 

Or,  41  dolls,  37  cts.  5  ms. 


PRACTICE* 

Case  2.    Page  C& 

cwt.    qr.         Ib.     Ib. 
«,      4         I  14=490 

1  =  122,5     Or,  122  dolls.  50  cts. 

civt.  qr.  Ib.     Ib.  cwt.  qr.  Ib.     Ib. 

S,       12    2    13  =  1413  4.     14     2    7=1631 

X  2  X  7 

Ib.                

3)2826  5x8=40)  11417 

S  942  8285,4*5 

Application. 
Page  70. 

cwt.     Ib.  hhds. 
Quest.  1.    12,5X112X6X^=3150  dollars. 

2.     60  ton  3.    12, 650  feet 

X  3  X  10 

5)180  126,500 

$isj  =     6.325 

tonEag.        36  fisj=     3,1625 

60X2=      120  |is^—     1,58125 


156  Eagles  137,56875 


Or,  1560  dolls.  Or,  137  dolls.  56  cts.  8|  ma. 

Case  3. 

dolls,  cts.  dolls,  cts, 

2.             10    94  3.           13    41 

Mult,  by         17  cwt.  Mult,  by        5  cwt. 

7658  67,05 

1094  1  qr.  is  i=3,352| 
f  qr,  is  one  hal  f           547 

1        is  one  half           g735  70,402J 

14  Ib.  is  one  half           13675  

4        is  one  seventh       3907  70  dolls.  40  cts.  2J  ms, 
I        is  one  fourth           976 

$19(5,0408 


Or,     196  d.  4  cts. 


PRACTICE. 
Page  70. 

dolls,    cts.  m§. 


105,035 

16  If),  is  4=  2.  14357+ 
£  lh.  is£  =  ,oG794  + 
1  Ib.  is  -£  =  ,13597  + 


107,58048  + 

107  dolls.  58  cts.&c. 
Case  4.    Page  71. 

s.  d.  s.  d  Or,  4f  3  yards 

6    8  is  I )  4*3  vds.  at  6  8  88|  cts. 


5784 
5784 

s.  d.  s.  d.  420| 

S;        S  4  is  J)  397yds.  at  3  4 

4 

Or,      597  yards 


1588 
1588 


Or,  159,25 
4.  1591/6.  at   15.  8r/. 


1850 


)  159  5  value  at  l£.       31850 
-  f=      3538| 


5  5 


S35,3888| 
Or,   S35  S8| 


67 


10*.  is  i 


Page  71. 
658  at  IS*. 


JE394  16 


Or,     658/Z>. 


39480 
658 

81052,80 


•12* 


6. 


745  at  16*. 

55.  is  5     372  10 

l5.i*-J  |  186    5 

37     5 


£j96     Q 


Or,    745  yards 


S  1589,331 


7.  Or, 

105.  is  i     069  at  195.  llrf.  969 


5  «  I 
4  '  i 
8d.'  i 


£964  19  3 


^tO'-t  1U 

242  5 

4845 

193  16 

5814 

32  6 

1938 

8  1  6 

£=    5S8J 

4r\  n 

u  y 

R9i7S  9..-;* 

55.   i 

3715  at  95.  4iJ. 

Or,  3715 

4  T 

P28  15 

—  —  —  « 

4d-  T\ 

743  0 

18575 

Gl  IS  4 

7^50 

7  14  91 

3715 

».  4lc 


£1741    8 


84643,73 


rRACTICB. 

Page  71. 

9. 

4567  at  195. 

ll|e/.  Or,  4567 

lOt. 

j 

_ 

2664 

5 

| 

2283  10 

4 

i 

1141  15 

27402 

Sd. 

i 

913  8 

£7402 

2 

| 

152  4  8 

9134 

1 
i  1 

i 

38  1  2 

10  0  7 

*=•=  507$ 

9  10  5| 

812153,29! 

•19*.  11  Id 


£4557    9 


Cas 

e  2.    Page  73 

. 

£.     5.     <J. 

A  «.  d 

«.       89     6     8X1 

S. 

8  11    5 

6 

9 

536     0     0 

77    2    9 

6 

8 

ewt.  3216    0    0 

617    2    0 

10  is  i  89    6     8 

Sqr.isi 

4     5     8J 

4  *  1  44  13     4 
£qr.  «  |  17  17    4 
141b.  «  i     2    4     8 

1   '        t 

16  Ib      | 
8  *        i 

£    2  10| 
1     4     5|  + 
0  12     2|  + 

0  11     2 

£  '«        1 

0     3     0£  + 

1  '        1 

0      1       D^ 

£3370  13     £ 

£625  11  10 

£.    5.     rf. 

A.  R.   Fl 

4.         S     17    6 

, 

5.     476    5     28 

12 

£     3    75.  lid 

%     10     0 

5s.        is  | 

1428 

12 

£*.  6r7.  «  | 

119 



3d.   '  j- 

59  10 

-  558     0     0 

2r.    is    i 

9  18     4 

2?r.is|*  1  18    9 

1         <     i 

1  13  Hi 

14/6.  <  |    0    9     8£ 

r     '  4  o  4  10 

20^.   *     ^ 
8         '    | 

0  16  111 

8     5|  + 

3         A  3. 

£560  IS    SI 

4| 

£1619  11     14 


PRACTICE. 

Page  72. 


d.  c. 
C.      10,55 

640  acres 

42200 
6330 

2r.  is  4       5275 
0p.      *       13181 


86758,5931 
Or,  6758d.  59c. 


8. 


d.  cts. 
6,34 
12 


7603 
396| 


S76,476i 
Or,  76d.  47  c.  6|ra. 


18,50 
229  acrei 


16650 
570 
570 

2r.  is  4       925 
1.     '4       4625 
Wp  *   *        1156J 
8       4  J-          925 

84252,4564 


Or,  4252d.  45c. 


9. 

dolls. 
14 
17  cwt. 

10. 

J,    e. 

125,50 
16 

238 

200800 

%qr. 

is  4 

7 

£r.     is  4 

6275 

\  < 

<  4 

35 

1        «  4 

31375 

14/6. 

7  * 

<  4 

£      | 

175 
875 

ZQper.  *  $ 
5  *     <  i 

156874 
3921J 

<il    6 

<      1 

4^Ti 

g251,5624 
Or,  251d.56c,2|ra, 


S2121,734| 
Or,  2121d.7r3c4m.-f 


11. 


Zqr.  is 

1  *     * 
14/6.   « 


TARE  AXD  Tr.ET. 

Page  72. 

£.    s.    d. 

3     17    6 

5 


99    7    6 
5 

96    17  6  value  of  25  cwt. 

1     18  9     4       of    2qrs. 

0     19  41  «       of    1  cjr. 

9  8i  4  14  Ib. 


£100  5 


TRET. 


Page  73. 

«0/.  yr.  II).           Ib. 

No.   1.     2  I  25  tare  21 

4      2.     2  2  0     4      21 

*       3.     2  0  21     4      21 


Gross       7    0    IS  tare  G3  lb.«2  qr.  7  Ib. 
Tare  2      7 

Ncatcwt.  6    2    11 


torn,  is  I 


J.    c. 

12,50 
6 

75,00 
6,25 


882,47711 


ciot.  qr.  Ib.     Ib. 
16     1     3=1823  gross 
132  tare 

1691  neat 
Mult.    ,45  cts. 


TARE   AND  TRET. 
Page  74. 

cwt.  qr.  ib.          Ib.  d.  cts, 

4.     No.  1.    11  1    17  tare  112  16,80 

4     2.    12  2      0      4       74  22 


Gross    23  3    17   tare  186  33,60 

1861b.=  1  2    18  336 

16$.  is  |      2,40 

22  0     27  neat  8  '  £      1,20 

2  <  i          30 

bbl-cts,  d.cts.  1  <i          15 

18X450     =  81,00  —  - 

Ton.  Uxl20  d.=l  80,00  §373,65  cost 

>1,  00  gave 
8261,00  paid 


cwt.  qr.     Ib. 

5.     4     2     26 

12 


56     3       4  gross  450 


Qqr.  is  5       4,50 


Sib.  isT^  4     06+  14         6|        1,12J 

4       «  £    2     0       3+  7         4  |           56| 

6    0      9  tare  ' 

8456,341 
Cwt  50    2    23  neat 

cwt.     Ib.  cwt.   qr.         Ib. 

6.      6=672  gross  7.     120  3  =  13524  gross 

100  tare  177  tare 

572  13347 

jV5538  22  tret  5^=    513,346  tret 


Ib.  550  neat  12833,654 

,73  cents 

Ibs.  cts.  Ib.        CO  5f)        cts- 

As  550  :  8250  ::  1  :  ~  =15     38500962 

89835578 


9368,56742 
Or,  9368  dolls.  56  cts.  7  ms. 

6 


62                                             SIMPLE    INTEREST 

Page  74. 

cwt. 

qr. 

ft. 

8. 

6 

12 

16    dollars 

9 

50  cwt 

59 

1 

2  4  gross 

800 

16ft.  is 

*~ 

1 

27 

\qr.  is  4 
14ft.     4   | 

4 

2 

1        « 

TV 

2 

3 

7        *   5 

1 

i     '  j- 

0,14f 

9 

0 

2  tare 

< 

£807.142. 

Cwt.   50     1     22  neat 

/6.     cict.qr.lbs      bags  Ib. 
9.    325(=2  3  17)   X27=8775  gross 
877aft.XTVf          =1018  tare 

7757 
7757-7-26=  298  tret 

7459  lb.=66c«tf .  2^r.   1 1ft*. 

Ibs.        ct.        dolls,  cts. 

And  7459X32=2386,88 
SIMPLE  INTEREST. 

Case  1.    Page  75. 

dolls,  dolls,  yr.    cts. 

4.  200x6x2=2400=24  dollars 

Page  76. 

doUs.  dolls,  yr.     cts.        dolls,  cts. 

5.  1260X7X4=35280=352,80 

dolls. dolls,  yrs.    cts.        dotts.  cts. 

6.  560X8X3=13440=134,40 

d.    cts.        d.        d.    cts. 
Now  1S4,40  +  560=*694,40 

d.         di    yrs.    cts.  cts. 

7.  4520X5X6=135600=1356,00 

d.     cts.        d. 
And  1356,00+4520=5876  dollar! 


SIMPLE    INTEREST.  63 

Page  76. 

dolls,  d.  yrs.  cts.  d.      d.       d. 

8.      400X6^X2=5200=52  dolls.     And  52+400=452 


cts.  ms. 
5 


dolls,  dott&  y-rs.      cents  d.        cts. 

9.  4925x7sX9=332437;|=3324  37 

dolls,  dolls.        cts.        dolls,  cts. 

10,  2500X71  =  19375=193  75 

d.        c.        d,  d.       cts. 

And  193    75+2500=2693  75 


Case  2.  Page  77. 

dolls,  c.  ms.  d.   cts.  ms. 

4.     1260  50  5  Now  283  61  3  + 

71  1260  50  5 


8823535  §1544  11  8  amount 

6302525 


cents  9453,7875  interest  for  1  year 


28361,3625 
Or,  283  dolls.  61  cts.  3+ms.  interest  for  three  years 

dolls,  cts.  dollars  cts.       d.    yrs.      d.        cts.ms. 

5.     630  50         6.     3462  84   X3^x4=450     169.2  ink 
8  1274646X73X3=286     79535  int. 


50  4400  int.  for  1  year  163     37385  cliff. 

5 


232  20  int.  for  5  years 

650  50  principal  dolls,  cts.  ms. 

Or,    163    37  3,85 
g882  70  amount 


7.     3  )  3422  25 

dolls,  cts.  ms. 

1140  75'3Tf  0=68      44    5  int.  for  1  year 
68  44  5  interest  for  one  year 

1209  19  5  first  payment. 


64  SIMPLE   INTEREST* 


Question  seven  continuedv 

dolls,  cts.  dolls,  cts, 

d.c.  m.  yr.  1140  75  d.  c.  m.        1140  75 

68,44,5X2=136  8  9  int.  for  2  yrs.    68,44,5x3=  205  335 

31277  64  2d  payment         3d  pay't  1346  085 

d.    c.  ms» 

Now,  1209  19  5  first  payment 
1277  64  0  second     < 
1346  08  5  third        < 


$3832  92  0  whole  amount 

Case  3.    Page  7S. 

4.         18  months  Then,  724  dollars 

4  per  cent  6 

12  )  72  S43.44  interest 

—  4,00  principal 

6  rate  for  the  time 

S767  44  amount 

5.         15  months  d.c.  m. 

7k  per  cent  Then,   9  375 

0  dollars 

12)11-1 

375000 

9,375  rate  for  the  time         18750 

22,50000 
Or,  22  dollars  50  cents. 

6,         4  me  dolls. 

Gi  per  cent.  Then,    1260 

12)26 

£[  rate  for  the  time  210 

30  cents. 


'SIMPLE    INTEREST.  65 


Case  4.     Page  80. 

dolls,     'days    (  const  v  !  div.)  d.       cts. 

5.  (12000X40)  ~-  52K  =  92    06  nearlj 

dolls,    days      (c.  d)       d.      c?s..  ms. 

6,  (8400X2p)-7-73CO=23     01     3 

d.    cts,    days     (c.  d.) 
r:     (517  90X8  l)-r-6083=7  dolls.  15  cts.  1m. 

d.    c£s.    days   (c.  d.) 
8.     (73  41X27)-7-GO=33cts 


.  cfc.    days     i'c.  d  )      d    c.   m. 

9.  (225  24X40)-r-{>083=l  48  1 

dolls,     days     (c.  d)      d.      c.  m. 

10.  (1£QO;X80)^6083=15  78  1 

dollars  cts,    dc,ys       (:.  d.)        d.     cte.  m. 

11.  (2962  19X254)~608S=123  68  8 

d.      cf5.    dvjs      (/;  '')      d      c.   m. 

12.  (1733  97  Xl05i)  -7-6083=^9  07  5 

Case  5.     Page  81. 

i/?'5.     d.          d.         d. 
2.       (12X6)4100--  -=172     Then, 

dolls.        dolls,  O^^OvlDO        d"118- 

As  172  :  100  ::  2750  :  -'-= 


yrs.   d.        d.  d. 

3,         (5X6)  +  1CO--130  Then, 

dolls.         doll?.          dolls.        TOPvlfin      dolls. 

As  130  :  100  ::  728  :  -^—=560 
130 

2/rs.   d.  d.         d. 

4.        (4  XTijy-  100=130  Then, 

dolls.         dolls.  foil-.-.         IhCOv/lOO       dolls. 

As  130  :  100  ::  1638  :  i±L:l±_±=l260 

JL  oU 

i/r*.  d.  c£s.    d.    d.  c^. 
5.    (5X5  75)+  100=:  128  75  Then, 

doll*,    eta.        dolls.  ii->lb.          °OOOXlOO       do^s<      cta< 

As  128  75  :  100  ::  2000  :  ^^^--  =  1553  S9 


SIMPLE    INTEREST. 

^       Case  6.     Page  82. 

clolls.         dolls.  dolls.         ^0800     dolls. 

2.      858  amount     As  650  :  280  ::  100  :  — — -=32 
650  principal 

Then  52  d.~-4  yrs.=8  per  cent 
S208  interest 

dolh.  dolls.        115200    dolli, 

'52  araount     As  1600  :  1152  ::  100  :  — 5 =72 

1800  prin. 

SI  152  interest.     Then  72  dolls,  -r- 12  yrs.=6  dollars 

cts.  dolls.       dolls,  cts.        dolls.       Ogn on     dolls. 

4.  800,80  amount    As  640  :  220,80  ::  100  :  -    — =34| 
640300prin. 

8220,80  interest     Then  34|  d.  -r-6  yrs.=5|  per  cent, 

dolls.  dolls.  dolls.  d.  gin     doJk 

5,  20100  amount     As  12000  :  8100  ::  100  :  =67! 

12000  prin. 


S  8100  interest.    Then  67£  d,-^15  yrs.=4|  per  cent. 

Case  7.    Page  83. 

dolls.  dolls. 

2.    650  910  amount 

8  650  principal 


852,00  int.  for  1  year     $260  whole  interest 

dolls,     year         dolls.        2()0 

Then,  as  52  :  1  ::  260  :  -—=5  years 


(lotts.  dolls. 

>.     1600  2080  amount 

6  160Q  principal 

&96,00  int.  for  1  year        S  480  whole  interest 

dolls^   year         dolls.        4gQ 

Then,  as  95  :  1  ::  480  :  — =5  years 


SIMPLE  INTEREST* 
Case  8.     Page  83.  ' 

2.    640^.  3*.  6flf.=2845|  dollars 
5 


8142,26-i-  interest  for  1  year 
3 

426, 78^  interest  for  3  years 
2845, 22f  principal 

83272,000f  amount* 


£.  -s.  cl 
3.  1374  1  9 
rate  per  cent.  5-f 


£.    s.    d. 

77    5     10  in'st.  for  1  yn 

38  12     11     <      for  |  yr. 


I  is* 


6870    8     9        £11518      9  int.  for  1J  yr, 
687    0  10i 
171   15     2| 


£77,29    4  10 
20 

5.5,84 
12 


.  10,18 


Case  9.    Page  85- 


&       1820,  January. 
falls. 
182V-  April  1.     1 
\            paid  700 

Principal 
Mult,  by 

1800  dollars 
days 
455  equal  819000 

1822~\Jan.  1.     1100 
j)aid    400 

Mult. 

275  equal  302500 

700 
Jul}A  1,-  paid  500 

Mult. 

181  equal  102500 

\             200 
\           214  16  6 

Am6uat\due  $414  16  6 

Mult. 

184  equal    3680ft 

6,0;  1285000 
Interest  $  214,766 

*The  answer  givcA  with  the  question  is  obtained  by  recsoninj 
$4,44  to  be  a  pound  sterling,  instead  of  $4,44£ 


68  INSURANCE. 

COMPOUND  INTEEEST. 

Page  87. 

tabular  number  dolls.       d.       cts. 

3.  1,27628X1500  =  1914  42 

tabular  no.        d.  d.         c.    m. 

4.  2,54035X4500=11431   57  5  amount 

subtract    4500  principal 


S6931  57  5  interest 

tabular  no.    d.  d.        cts. 

5.  3,20713  X650=*2084    63  4,5  mills 

yrs.  yrs.- 

6.  ',  21 — 4=17  years,  on  interest     Then, 

tabular  no.     d.  d.        cts. 

2,69277X8000=21542  16  amount 

d.      cts.    sons 
But  21542  16-7-3=7180  dollars  72  cents 


INSURANCE. 

Page  88. 

dolls. 
2, 


dolls. 

doBs. 

1650 

3.        4500 

15|  per  cent. 

25    per  cent 

8250 

22500 

1650 

90 

825 

£ii0.*i.nn 

8255,75 
100  dollars — 25  dollars=75  doMars,    Then? 

dolls.        dolls.  dolls. 


.  .  . 

As    75  :  100  ::  4500  :  = 

i  5 

** 
100 dolls.— 9  dolls.  =91  doll?.     Thetf 

k  dolls,        dolls.  dolls.        560DO       dolls'        cls 

\Ag     91  :  100    :    560:  -—^=615      38$f 
y  JL 


4. 


BROKAGE. 

COMMISSION, 

Page  88, 


dolls. 
1260 

6  per  cent. 


875,60 

dolls. 
26342 

3  per  cent, 


§790,26 


Page  89. 


100  dollars. 


As  103  :  100  ::  4120 


dolls. 

2550 

4  per  cent 


5. 


8102,00 

dolls. 
6422 

3  per  cent. 


4)19266 


41^000 
~' 


=4(100  dolls, 


dolls,  hir          dollars         4000 

Then,  as  8   j  1  :  :  4000  :  ---  =500  barrels 


d.  cts. 
.50 
3}  percent 

4876,50 
541,83| 

5418,33 
Or,  54  dolls.   18]  cts. 


3. 


dolls. 
1868 

\  per  cent 

3736 
934 


846,70 


4. 


560  dollars 
6  per  cent 


833,60  cents 


TO  REBATE    OR     DISCOUNT. 

BUYING  AND  SELLING  STOCKS 

Page  90. 


dolls. 
2.     1500 
110  per 

dolls. 
Or,   1500 
cent.  -^0  =  150 

dolls. 
3.     1686 
X     128  per  cent. 

81650,00 

dolls. 
4.        25000 
108 

S1650 
per  cent. 

13488 
3372 
1686 

82158,08 

827000 

d.  d.  d. 

5.      1260                6.  9254                7.  1518 

90  per  cent.  84  per  cent.  S3?  per  cent 


81124  S7773,36 

REBATE  OR  DISCOUNT. 

Page  91. 

mo.     d.         mo.      d. 
2.       As  12  :  8  ::  18  :  12  int.  for  the  time      Then, 

d.         d.  d.        d.  2464x100      *L»- 

As  112  :   100  ::  2464  :  2200  Because—  ^^  =2200 

mo.      d.          mo.      d. 
S.        As    12   :-5i   ::  8   :  3  f  interest  for  the  time 

d.  d      cts.         d.       cts. 

Then,  aslOSf   :    100  ::   1857  50   :    1791  80/T°T 

1  85750X100X3      ^   cts' 
For   -  rr          -  =1/91 


4.        (7X2)  +  100=114  dollars 

doiu.     doiis.        doiis.      650X14    *£V£. 
Then,  as  114  :  14   ::    650  :  =79  82|4 


REBATE    OR    DISCOUNT.  71 

Page  9J. 

mo.      mo.       d.      d. 
5.     As  12  :  4f  ::  8  :  3=  a  rate  for  the  time 

dolls.         dolls.  dolls.          5150X100        d°N9- 

Then,  as  103  :  100  ::  5150  :  -  -  — —  =5000  p.  w. 

lUo> 
Again, 

dolls.        dolls. 
As  100  :  99  (=100—1)  ::  5000  :  4950 

Page  92.     Note  1st. 

mo.      mo.     dolls,  dolls. 
3.    As  12  :  18  ::  8  :  12  rate  for  the  time 

dolls,     dolls.       dolls.      dolls. 
Then,  as  112  :  12  ::  7280  :  780  the  discount 

dolls,    mo.     cents 

Again,  7280x9=65520  interest  at  6  per  cent. 
i =21840  interest  at  2    <       * 

$873,60  interest  at  8  per  cent, 
Subtract  g780,00  discount 


Difference  8  93.60 

Note  2d, 
dolls.  dolls. 

%.        1650X3=4950=-49  dollars  50  cents. 

3.      2464  dollarsx8=~19712  cents  discount 
But  2464  dollars— 19712  cents=2266  dollars  88  cents 

BANK  DISCOUNT. 

+  Page  93. 

days  dolls. 

2.       — =1  dollar  41-f  cents 

60 

Or,  17X25°=1  dollar  4 If  cents 

days  dolls. 

3.  (90+?X600=9  dollars  40  cents 

ou 

Or,  47daysx600dolls=9  dollars  40  cent. 
30 


72  SINGLE    FELLOWSHIP. 

Page  93. 

days  dolls,    cis. 

(604-4)  XI  260  40      f_  ,  „ 

4.  v  -  1—  i-  -  =  13  dollars  44||  cents. 

uu 

~      32X1260  40 

Or,  -  -  --  =13  dollars  44ff  cents. 

days  dolls,  cts. 

(60-f4)x2649  75     rt0  ,  :,  ... 

5.  i  --  £  -  =28  dollars  26  cents  4  mills 

60 


Or,  =28  dollars  26  cents  4  mills 

bO 

EQUATION  OF  PAYMENTS. 

Page  94. 

dolls,  mo.  do/Zs.      mo. 

2.     500X2=   1000  3.  £OOX  5=1000 

1000x5=  5000  SOOX  8=2400 

1500X8=12000  500X10=5000 

-  -  -  --  mo.  da. 

3000    )     18000  (6  mo.  1000    )      8400(812 
18000 

4.     1400  dollars—  1000  dollars=400  dollars 

d  d.         mo. 

Then,  as  400  :  1000  ::  1  :  2J  months 

SINGLE  FELLOWSHIP. 

Page  95. 

d.  d.  d. 

2.    A  7500+B  6000+C  4500=18000,  Then, 

dolls.        dolls. 

dolls,     dolls.  f7500  :  2250  A*s  gain 

As  18000  :  5400  •  .  •  10  :  3  :  :  <  6000 

(_4500 

d.      cts.  d.     cts. 

5.     A  1291  23+B500  37+C  709  40+228=2729 

d.  d.      cts,          d.          d. 

Now,  as  2729  :  2046  75  ::  JOO  :  75 

d.  cts.   d.  cts. 

:  963 

Then,  as  100  :  75  ::  \  500  37  :  375 
Or,    4:3  ::  ^  709  40  :  532  05 
171 


dolls. 

:  2250  A*s  gain  "] 

:  1800  B's     4  I 

:  1350-  C's     *  J 


d.  cts. 

(  1291  23  : 
:  75  ::  \  500  37  : 
:  3  : :  \  709  40  : 

/  228  00  : 


•SINGLE  FELLOWSHIP.  73 

Page  95. 

dolls,    J.  R.  P. 

dolls.    A.  R.  P.        f  ISO  :  175  2  34j  1 
-4L     As  600  :  585  2  34  ::   <  195  :  190  1  17^  > 

(.225  :  £19  2  22|  J 

Page  96. 

barrel*. 

IU.   bbl.          f  960  :  4001 
5.  As  2160  :  900  v  12  5  :  :  <  720  :  300  [ 

^480  :  200  J 

dolls.         d.        d.  dolls,     d.       d. 

6,    A  1260+B  840=2100,        And  825—275=550 

d.          d. 

d-         d.          C  1260  :  330  A'se;aia? 

Now,  as  2100  :  550  :  :  <  C 

£    840  :  220  B's     '     5 

B's  gain.  B's  stock.  d.  d. 

And,  as  220  :  840  v    11  :  42  :  :  275  :  1050  C's  stock 

dolls.        dolls,  dolls. 
7.    (80dx£)-f  40=1640.  whole  gain,  And 

d.  d.  d.        d. 

A's  140-f  B's  260-fC's  300=700 

Then,  800  dollars—  -700  dollars=100  dollars  D's  stock. 

dolls.      dolls 

dolls.     Mis  K 

Now,  as  800:  1640  S  : 

LlOO  :  205  D's 


8,  d.   cts. 

cattle  dolls. 
As  SOO  :  96  v  25 


«.     CIS. 

f   80  :  25  60  A  pays  1 

.  R         <  100  :  32  00  B     4  L 

'    (.120  :  38  40  C     4  J 


9.     First  2+3+5=10  dollars,  Then 

ddlh. 

r  /*    . 

dolls,  dolls. 
AB  10  :  5000  •.•  1  :  500 


f2  :  1000  A  gets") 
-!  3  :  1500  B    <      {> 
'    15  :  2500  C     <    J 


I  COMPOUND   FELLOWSHIP. 

Page  95. 

10.    20  dollars+85  dollars=105  dollars  whole  rtock 

d.       d.      d.     d. 
Now,  as  105  :  63  ::  20  :  12  A'sgain 

d.       d.     d.  d. 

And  63— (21  +  12)=63— 33=30  C?s  gain 

d.      d.    . 

doii,.  ton*.      T21  :35Bputin  7 

Then,  as  12  :  20  ::    2  C 

£30  :  50  C  put  in  } 

COMPOUND  FELLOWSHIP. 

Page  97. 

d.     mo.  d.      mo, 

2.     600X8=4800  1200x10=12000 

200  600 

800x8=6400  600X   6=  360D 


A  11200  B  15600 

1000X12=12000  11200  A 

800  15600  B 

. 19200  C 

1800X  4=  7200 

460CO 

C 19200 

dolls,     f  112CO  :  560  dolls.  A>«  sham 
Now,  as  46000  :  2300  ::<  15600  :  780  dolls.  B's    * 

That  is     '20      :       1  [  19200  :  960  dolls.  C'i    4 

mo.    mo.     dolls,    dolls. 
3.        As  9    :   12  ::  600     :   800 

Page  98. 

mo.  mo.       d. 
4.      As  8  :  12  ::  1800  :  2700  doikrs 

mi.         mi.         ml,        mi. 
5.     A  600-f  B  600-f  C  300+  D  100=1800  miles 

T600  : 60  dolls.  A  pajs 
milts    dolls.  j  fiftn .  £0     <      R     * 

As  1600  :  160-,'    10  :  1   ••SsooJsQ     «      C     * 

I  100:  10     *      D     * 


PROFIT  AND   LOSS,  75 

Page  98. 

5  oxenx4£  months  =22i 
8          X5  =40 


Sum  121 

<J.   CJS, 

d.  cts.   f  22J  :  27    A  pays 
Now,  as  121  :  145  20  ::  <  40  :  43    B   * 

[58|  :  70  20  C      « 

r.  4000  dollars  Xl2  months=48000 
SOOO  «  X15  '  =45000 
5000  «  X  8  '  =40000 

133000 

r/.         f  48000  :  240  dolls.  A's  part 
As  133000  :  G65  ::  <  45000  :  225      *      B's     * 
That  is  200  :     1          [40000  :  2CO      <      C's     * 

8.  580  dollars  x6J  months  =3770 
580  -*  x9k  6  =^5510 
870  «  X8f  '  =7540 

16820 

d.  cts.   ("3770  :  59d.  15c.  A  lost 
As  16820  :  263  90  ::  <  5510  :  86  45   B  * 
[7540-,  118   30   C  * 

PZOF1T  AND  LOSS. 

Page  99. 

4.  150  cents,  sold  for 
120  cents,  cost 

30  cents,  gain  on  120  cents 

cf^     cts.        (].       d. 
Then,  as  120  :  30  i:  100  :  25  per  cent 

u 

yd.   cts.     yds.     d.  cts. 

5,  As  1  :  66  :  :  42  :  27  72  sold  for 

Subtract  21  00  cost 

Rem.     S  6  72  gain 


*  O  .  PROFIT  AND  LOSS, 

Page  99. 

4.  52  galls.xG  barrels=192  gallons,  bought 

Subtract     12  gallons  leaked  out 

Rem.     180  gallons 
And  96dollars  +  12  dollars=108  dollars  sold  for 

Sails.         d.        gal.        cts. 

Then,  as  192   ;   96  .•:   1    :    50  cost  per  gallon 

£.        d.  g.        cts. 

And,  as  180:  108  ;:  1   :   60  sold  for  per  gallon 

5.  20  cents — 17  cents=3  cents,  loss  on  each  knife 

knife    cts.  knives      doz.      d.   cts. 
As    1   :    3  ::    1440  (=120)  :  43  20 

Page  100. 

149  dollars  7.-  100  dollars 

51     "  18     " 

yds.     yd.  d.  d.        —    cts,   d.  c.  m& 

As  100  :  8200  :;  'l  :  2  As  100  :  S82::  125 :  1  02  5 

d.      d.    cts. 

8.  As  100  :   120  .*:  90  :  1    08  must  sell  for 

108  cts.— 90  cts.  =  18  cts.  gain  per  hat 

hat     cts        hats         doz.        d   cts. 
Now,  as   1  :  18  ::  240  (=20)  :  43  20 

9.  115  gallons.XllO  cente=126  50  cost 

Add  5  00 


S131  50  must;sell  for 

But,  as  1  doll.  :  1  gall.  :.- 131  dolls.  50  cts. ;  131|  galls/ 
the  number  of  gallons  that  must  be  sold.    Consequently 
131|  gallons — 115  gallons=l65  gallons  of  water. 

dolls,  cts. 
10  Ib.        ciot.  cts.     134  40  cost 

448    (=4)X25=  112  00  sold  for 

g  22  40  whole  loss 

ti>.          doiis.  cts.       U>. 
Then,  as  44S    .-   22  40  :;•  1  .'  5  cents  per  Ib, 


w 

BARTER.  77 

Page  100. 

dolls.       dolls. 
11.    As  120  :  100  ::  17-28  :  1440  whole  cost 

yards.       dolls.       yd. 
Then,  as  360  :  1440  : :  1  :  4  dollars 

1&    84  90—84=90  cents;  Then, 

dolls,    cts.          dolls.       dolls. 

As  4  :  90  :  :  1000  :  225 

d.  d.  cts. 

13.  As  120  :  I    ::  150    :    1  25  per  bushel 

d.  cts.    lu.        d    cts. 
Now,   as  1  25  :  1   ::  57  50  :  SO  bushels 

14.  5000  galls,  at  48  cents=2400  dellars 
Interest  for  2  months  =     32     ' 


Costs  82432  to  pay  cash 
5000  galls,  at  50  cents =82500  in  two  months 

Diff.     8     68  gamed  by  borrowing 
the  money  and  paying  the  cash* 

head  dolls,    dolls. 
15,     12 )  240  (  20  the  cost  per  head 

cts.    Ib.       dolls.       Ib.        C.qr.lb. 
Then,  as  4  :  1  : :   20  :   500=4  1  24 

BAR  TEN. 

Page  101. 

2.     320  bushelsxgl  20=g384 

Subtract    160  cash 

.  8224 
cts.     Ib. 
Then,  as  20  :  1  : :  224  dollars  :  11201k 

cts.      cts.         bu. 

3,  As    70  :  125  : :  28  :  50  bushels 

cts.       cts.  m.          cts. 

4,  As  250  :  319  : :  23i  :  29  9|£,  or  30  nearly 

d    Ct8,          ft    Ct9, 

5,  As  100   :  120  : :  1  25  :    1  50 


EXCHANGE* 
Page  102. 
d.    rU.       d.    ct».        d.    cts,  o  gf)  y  1  Ort      4-    ct*.    auu 

As  2  50  :  2  80  ::  1  20    :     —  c  *  *~  =1  34    4 

cts.     cts.         bu. 

As  99  :  90  :  :  240  :  218S*T  bushels 

ets.       cts.       cts.     d.    cts. 
And,  as  90   :  95  ::  99   :   1  04  5  mills 

oz.       if.  cts.  oz.  Ib.   oz. 

As    16    :   1  28  ::  420  (=26  4)  :  S3  dolls.  60  cts  : 

cts.  Ib.     d.  cts.    Ib. 
Then,  as  6  :  1  :  :  S3  60   :  560=5  cwt 

eta.        cts.  eta. 

9.     As   22  :  25  :  :  28  : 


110X31    gr         Ctfc 

Then,  as  100  :  110  :  :  31TflT  cts.   :  -  ^=35 

10.    250  barrelsx6  dolls.  25  cts.  =1562  dolls.  50  cts. 

Ib.      cwt.        cts.  dolls,  cts. 

And  448  (=4)    at    30=134   40 

64  Ib.  at  gl   75  =  112   00 

^25  7.  at  g6        =150   00 

Cash  206   10 

S602  50 

do  11$.  cts.    dslls.cts.    dolls.  d.     bbl.         d        6W. 

;  562  50—602  50=960    Then,  as  8  :  1  :  :  960  :  120 


EXCHANGE. 
Case   1.      Page  104, 

d.        8.        d.        s.  d.        £.        £. 

96  (=8)  :   90    (=T  6)  : :  256  :  240 
Or,     £256 

^=16  subtract 

8240 


EXCHANGE. 


Page    104. 

d.          s.    d.          d.          B.  d.         L.          L.    *.    d, 

3.     As   90  (=7  6)  :   56   (=1  8)::  1500:  933  6  8 


^  •     '     . 

^1=933    6   8    Answer  as  before 


D 

g.    s.      £.        £. 

4.  As  6  :  8  ;:  180  :  240        Or,      £.180 

i-  =  60 

£240  as  befor® 

d.        s.  d.        d.        *        £.       £. 

5.  As  56  (=4   8)   ;  72   (=6)  ::280  :  360 

Or,     £.280 
f  =  80  add 

£360  as  before 

d.        s.        d.        s.  d.          £.        £.  s.  d. 
6,    As   72   (=6)   .•   56  (=4  8)   ;:   304  ;  236  8  10| 
Or,     304 

£=  S3  15  6f 


270     4  5k 
33  15  6-| 

£236     8  lOf 
Case  2.    Page  105, 

£.     s.  d.  shillings  s.          s.     dolls,  cffo 

4,  468  19  6=9379,5      Then,    9379,5—6=1563  25 

£.    s.      pence  d.        d.     dolls,  cts. 

5.  750  10=180120    Then,  180120-7-90=2001  33{ 

Case  3.    Page  108. 

.£      x  ® 

9,    540  15=540,75     Then, 

£540,75X84,10=32217  07i 


6. 


4  dolls,  1.0  cts 


ALLIGATION 

Page  108. 

£.     cl  cfs.     d.    cts. 

7.  320  X  4  44  =  14-20  80  at  par 

12£  is£  =  177  60  add 

S139S40 

Page  109. 

8.  S3259—S4  10=^794,873,    Then, 

£.  £.  £. 

As  108  :  100  ::  79-1,8:8    :    735,9999,  or  736  nearlj 
Otherwise, 

£•     S.    (L 


34,10X108 

L  1  0  7               "                          3< 

ALLIGATION. 

£.       20  Ihs, 

so  *• 

40  " 

Case  1. 

,  at  10  cents—  £00 
at  15      "     =  450 
at2J      ;i     =1000 

90  cts.  1650 

cte.      /^.   c^c 

Now,  as  90  :  1630  ::  1  :  18| 

3.     10  bushels  at  150  cents=-1500 

20      ^       M    160     ••     =-3^00 
SO       "       at  170     "    =5100 

60  9SOO 

bu.     d.      bu.      cts. 
Now,  as  60  :  9S  ::  1   .-  16S| 

P^.ge   110. 

4»     4  ounces  at  7    c^nts=SOO 
8       "        at  CO    "     =480 

12  780 

02.        r'y        OZ.        Ct3+ 

Now,  as  12  .•  780  :t  1    :    65 


ALLIGATION. 
Case  2.     Page  110. 

gallons  at  3  dollars 


("3\\   1+2  —  3  gallons  at  3  dolla 
K5/)  i       =.1     '    «      at  5       4 
(^  '  1       =1        *      atG      ' 


40  5 

5 
20//  15  +  5=20 


Or,    Or, 
1 

1 
4 


6  bushels  of  rje 

6        *       of  corn 

24        6       of  oats 


x| 

/1 


iuo) 


Cfe 


4.    1st.  f!20  )20.at.l20M&ri20\   1 10+20=30  at  120 


110\|10       110|inn!  110A  I  =20      110 


ul    90/110        90 
L  80  J20         80 


100 


I 


90    )|  =20        90 

80  /J  20+10=30        80 

f 

Ci5.  Cf4T. 

3d.     f!20v    lOat'UOU/A.    (  120  N  ±=10atl20 


Iix-L/v      1 1/  ui    i -cu  i  Tt/t«     i    i  -oiy    »  ~^=ii/ai>  ix»u 

110)v20  110J  inn  )  110\)X10  +  20^=30  110 
90;)20  90  j  JJ  90//)20+10=f=SO  90 
so  /in  80 1  C  80  '  =plO  80 

10  +  20=30  at  120  cents 

=  10  110 

120+10=30  90 

=20  80 

120     N  =20  at  120  cents 

10\\  |10+20=30        110 
90/1  I  =10          90 

/20  + 10=30          80 

10+20=30  at  120  centa 


120+10=30       90 

y20  + 10=30          80 
Case  3.     Page  112. 


ass    4 

'  \  w  =4 

~  )-30\)| 

26+14  +  8=48  given, 

',  as  48  :  24bu  : :  4  :  2  bushels  of  each  sort 


H 


REDUCTION   OF  VULGAR   FRACTION3. 
Page  112. 


S.  /    0 

\16 

22  <  18 

/  20 
*  24 


SB  2 

r=  2 

=  2  quantity  girea 

=  2 


2  :  10  Ib.  of  alloy 

2  :  10       of  16 

2  :  10       of  20 

t34  :  170       of  24 


Cue  4. 


2. 


2 


16 


• 


2  :    5  oz.  of  15  carats  fin* 
The*,  «  16:  40::<     J;    *       «f  JJ     |        ,' 
O  :  25        of  29       *         > 

-Cr  ^0/5.      C  5  58s  galls,  of  wine 

^0/1   Then,  as  6  :  70  ::    ^1  llf  galU.  ofwiter 

a 

REDUCTION  OF  VULGAR  FRACTIONS. 
Case  1.    Page  114. 

Now,  72  )£&(=} 


8.       144)216(1 
144 

— 

torn,  measure  72)144(2 
144 

5.    75  )  125  (  1 
75 


(8)     (9) 
144    18 
Or,  --  =—  =*§  as  before 

216    27 

.    Then,  25  )  fa 


50)75(1  (5)     (5) 

Or,  f/T=if  =*f  as  before 


eoau  measure  25  )  50  ( 


OP  VULGAR  FRACTIOUS.  83 

Page  114, 


4.       4800)10800(2         Then,  1200  )• 
9600 


com.  measure    1200)'4800(4        Or,  T££ $%**=%  as  befora 
4800 

5.       91)117(1 
91 

Then,  13)TVr.(«l 
26)91(3 
78 

com.  measure    13)  26  (  2 
26 

5.    9876)88884(9  Therefore,  9876 )  ¥\V¥6y  (t=|  • 

88884 

Case  2.     Page  115. 

12x17-1-15 

2.     ! =2Ty>  fraction  required 

183x214-5 

3.     — =  If-8  fraction  required 

514x164-5     8229       '' 

4.    _ 3=3B***»  traction  required 

'   _      68425x443   '    .    .03  r 

5.  _^ —   -=.27^70,    fraction  required  . 

Case  3. x  ' 

%.  3848-~21=:18325r  equivalent  mixed  number 

3.  2465-~7=352|  equivalent  mixed  number 

4.  96 1-7-17= 56 f9T  equivalent  mixed  number 

5.  8229-7-16=514/5  equivalent  mixed  number 

Case  4.     Page  116. 

ft.       3x5x6=  90  first  numerator 

4x4x6=  96  second  numerator 

5x4x5  =  100  third  numerator 
And  4x5x6=120  common  denominator 

Therefore,  T\<^,  ^  and  {fg,  are  the  fractioni  reqmrtd 


KIDUCTION   OF   VULGAR    FRACTIONS, 


5. 


Page  116. 

1X5X15X  9=  675  first  numerator 
SX3X15X  9=1215  second      ' 
4X3X   5X   9=   540  third          « 
5XoX   5X15  =  11*25  fourth        * 
And  5x5xl5x  9=2025  common  denominator 
Therefore,  -2V/5-,  |§li?  &%  and  -Jiff  are  the  frac.  required 

Case  5. 
2.         4)3     8     13     20 

5)3     2         35 


1215 

How,  4x3x2x5=120  common  denominator 
Then,  1 20  com.  denominator 


8 
Divisors  <  j0 

L20 
Hence, 


5. 


40x2=80") 

15X5=45  I   XT 

10X4=40  ^ew  numerator* 

6x5=50j 

45          <10      ovi/1      30 

T20'  T20  "I1U  T35" 
Page  117. 

3)3     5     15     9 
5)1     553 


1113 

Now,  3x5x3=45  common  denominator 
Then,  45   com.  denominator 


Divisors  <|  J 

19 
Her 

15X1  =  15") 
9X3=27  ! 

5x5=25j 

ice   —-   -  7    -1  2 

New  nu 

Case  6. 

S.      7X4X  9=252  num. 
8x6x10=480  denom. 

Or,  3 
7x4x# 

7x5 


Hence, 


REDUCTION  OF  VULGAR  FRACTIONS.         85 

Page  117. 
3.     5x4x3=  60  num.  5x^X^_       5 

9X8X4=288  denom.       *'  #    $    4~~3x2x4  ^T 

ffpnrp      60   _   5_  32 

Ai.CIlL"j  -s-o  ~o  ~~"'2'X 

4.  5x4x11=220 num.  5x£>Cl'l       5x11 

9x7x12=756  denom.         '9     7    £$     9x7x3~~f^^ 
tlencej  Y5"s"==Y'^"5 

Case  7. 
4X1X1 
5x"20xT2==7"2~^==3^  fraction  required 


Page  118. 
/10X1X1  1 


4. 

11X 

6 

Case  8. 

1X12X20  A,      .. 

2'     30^:^===!^-=^  fraction  required 

1x63x8 

3.     -ITXT; =T?|=TI  fraction  required 

728xlXl 

1 X  24x60 

=T|||=="TT  fraction  required' 


Case  9. 

2.        12  ounces=l  pound 
3 

5)36 

7  ounces  4  pennyweights 

Page  119. 

3.     8  furlongs=l  mile  4.     24  hours^l  day 

4 

5)32  10)72 

6  furlongs  16  perches  7  hours  12  min. 

8 


35  ADDITION    OF   VULGAR   FRACTIONS. 

Page  119. 

5.        100  cents=l  dollar 
4 

5)400 

80  cents 

Case  10. 

3,        31  galls.  2  qts.  63  galls,= 1  hhd. 

4  4 

126  quarts  252  quarts 

Wherefore,  v||=l  is  the  fraction  required 

cut.  qr.  Ib.  cwt. 

4.     6  2  18f-  20=1  ton 

4  4 

26  80 

28  28 

216  £240 

53  5 

746  6720  thirds  of  a  pound 


2240  thirds  of  a  pound 
Wherefore,  ff f£=  J  is  tne  fraction  required 

Case  11.     Page  120. 
g.         20)  17,00  (,85 

ADDITION  OF  VULGAR  FRACTIONS. 

Case  1. 


25 

15+25+45+55_140  _ 
60  *°" 


ADDITION    OF   VULGAR   FRACTIONS*^  87 

Case  2.    Page  121. 
Given  denominators  40  com.  denominator 


2. 

2)2  458 
2)1  2  5  4 
1152 

And  2 

5 
8 

20  first  numerator 
10  second     4 
8  third         < 
5  fourth      ' 

Then,  2x2x5x2=40  common  denominator 

20+10+8  +  5 

Wherefore,  — =  14= 1*3- 

40 

Given  denominators  840  com.  denominator 

3.     5)567815 


Then,  5 
3)16783  6 


2)1  2  7  8     1 


15 


168X4=672 
140x5=700 
120X6=720 
105X3=315 
56x8=443 


11741  - 

Now,  5X3X2X7X4=*840  com.  denoni    2855  sum 

Wherefore,  ^ 


Case  3.    Page  122. 
3.  oH= 


Wherefore, 

4.        §of|  =  ^ 

TK.      tT-..'!     108  +  105+40+60 
Then, 


Lastly, 

Case  4. 

ft.  in.  h.  m. 

2.     £  of  a  yard=  16  3.     I  of  a  day==  8    0 

|  of  a  foot  =08  i  of  an  hour=0  30 

Sum  2ft.  2  in.  Sum  8  h.  30  mi. 


8S  MULTIPLICATION    OF   VULGAR   FRACTIONS. 

Page  122. 

d.  h.  m  yds.  ft.  in* 

4.    £ofaweek=28    0  5.   |  of  a  mile=1540  0  0 

k  of  a  day    =060  fofayard=       020 

i-  of  an  hour— 0  0  50  $  of  a  foot  =      009 


Sum  2  d.  14  hrs.  SO  mi.  1540y.  2ft  9in> 

SUBTRACTION  OF  VULGAR  FRACTIONS 

Page   123. 
1  1  _  q 

4          11        J5_-1  1          9  _  A1         y  _  2   _  1 

T?  —  4  —  T  i  —  1  2  —        0  ~  —  i~2  —  5* 


05  _  04 

—  . 


—  .-"IS  —  M^ 


6      MJ^j    .-..-I.       ..,_418-gl_ 

r-T—  ^'32      '.  —  4T3T 


8.  l£-/5-6|  =  12^—  6-^=5-}| 

9.  13i—  8if=13/f—  8if=- 

10.  10£—  1-&=IOJ{[—  U&=5 

11  1Q5  7    _  1O75  77    _  1Q163 

11.  iyri  —  TJ^iOT6^  —  Tg-T  —  ioT6T 

/.    /i.   ??iz. 

13.        7  weeks=7    000 
9-^days=l     2  16  48 

Diff.  5w.  4d.  7h.  12  min. 
MULTIPLICATION  OF  VULGAR  FRACTIONS. 

Page  124. 

4  _  12  _  3 
-"40—  TO" 


5,  ^yx^V^ 

6.  I  of  9=V     Then,  2J  X| 


RULE  OF  THREE  IN  VULGAR  FRACTIONS.  89 

Page  124. 


DIVISION  OF  VULGAR  FRACTIONS. 

Page  125. 

5.    H-^4T8o-|xA=M-==i56  6.    1-4=1x1=^ 

8.     5205^-=26f26,  and  f  of  91  =  3£4 

RULE  OF  THREE  IN  VULGAR  FRACTIONS. 
2.    As'f  ton.  :  4|3  dolls.  (=164£  dolls.)  :  f  ton. 

n 

X    493    0     1479     dolls-   ct3' 
Then,  ^X-Y-X-=— -=211  28| 

'19 

dolls. 

5.  }  of  $=25oS  Then,    As  ^  :  171  ::  1 


7 

yd.     dolls.        yds.        A$  v  ff    .  dolls. 

4.     As  1  :  f  ::  42  :  ^-—^=35 


5.     f  off-  A;  Then  say, 

dolls.  ' 

Ai*    :  312  ::  1  : 


Page  126. 

6.  ll=!,79!=2£8,  and3|=V7 
Now,  as|6z/.  :  2J7c/5.  ::  ^bu.   : 

^    238    17 
Because,  -gX—  Xy==4f^6=202rV 

7.  As  17535  :  |  ::  60120  :  | 

60120 


8* 


_)LUTIONV 

L\TE2!SE  PROPORTION. 

Page  126. 
.     61,cwt.—  \/cwt.  and  22^=^-1  miles 

cwi.  cwt.      mi.      miles 
-,  as  1   :   y  ::  ^y    • 


53  1 
;^r?*=-—  =145|  miles 


S.     As  16  :  12  ::   1|8  (=37|)   :  28}  days, 
47      3 


Because,  =  '|>=28j  days 

4 

4.     100|  do!ls.=  3f  2  dolls.  6f  mo.=\°  month:? 
5}  years=V  jears=46  months 

mo.    mo.     dolls. 
Then,  as  46  :  2T°  ::  3f2  :   14||f  dollars 


/<.  //.     /if.        /?. 
5.     Asl   :  I  ::  V  (=20i)  :  1/ 


Because,     XX==1f3=17|  feet 

/       J?4        P»  / 

6.     20|  yards^V  yards     1|  jard=f  yards 
Now,  as  |  yd.   :  J-  yd.  :  :  %l  yd.    :  34  J.  yards 

Because,  ^X^X-7=2|5=34-J-  yards 

i      2 

INVOLUTION,  OR  RAISING  OF  POWERS. 

Page  123. 

2.  (35)4  =  35X35XS5X35=1500625 

3.  l033  = 


SQUARE  ROOT.  91 

Page  128. 

4.     (,029)5=,029X,029X,029X,029X,029 
=,000000020511149* 

5.    l4=l 


SQUARE  ROOT. 

Page  129. 

2.    100929(327  root  required        3.    4762969(2187  root  required 
9  4 

62)169  41)78 

124  41 

647;  4529  428)3729 

4529  3424 

4367)30569 
30569 


4.  43046721(6561  root  required   5.  337420489(19683  root  req. 
36  1 

125)  704  29)237 
625  261 

1306)  7967  333)2642 
7836  2316 

13121)  13121  3928)  32604 
13121  31424 


39363)  118089 
118089 

6.    22071204(4698  root  req.     7.    36372961(6031  root  req. 
16  36 

86)  607  1203  03729 

516  3609 


929)  9112  12061)  12061 

8361  12061 


9388)  75104 
75104 
"The  answer  given  with  the  question  is  the  4th  power  of  ,0 


9&  SQUARE  ROOT. 

Page  129. 

8.    2268741  (1506,23+  root  req.  9.    7596796(2756,228-r-root  req. 
1  4 

25)  126  47;  359 
125  329 

3006J  18741  545;  3067 
18036  2725 

30122 )  70500  5506;  34296 
60244  33036 


301243;  1025600  55122;  126000 

903729  110244 


121871  remainder          551242;  1575600 
1102484 

5512448)  47311600 
44099584 
Page  130.  *?          3^12016  remainder 

10.  9712,718051(98,55^1001  req.    11.  3,17218120(1,78106+1001  req 
81  1 

188)1612  27)217 

1504  189 

1965)10871  348)  2821 

9825  2784 

19705; 104680  3561)   3781 

98525  3561 


197103;  615551  356206)  2202000 

591309  2137236 

24242  remainder  64764  remainder 

15.  4795,2573l6;69,247-Hootreq.     13.  ,00008836 f,0094  root  req. 
36  81 

129; 1195  184;  736 

1161  736 

1382;  3425 
2764 

13844;  66173 
55376 

138487;  1079710 
969409 

110301 


HOOT. 


93 


Square  root  of  vulgar  fractions, 

Page  130. 

1.  The  greatest  common  divisor  is  576 

«TA\2304  -  4.        ThATl      A/4  -  2 

S'OJ-STFT  —  "9?          en?  v  -3  —  "3 

2.  The  greatest  common  measure  is  169 

)!l§t^f;  Now,  v^f=* 


46656(216  denom, 
4 

41)066 
41 

426)2556 
2556 
,  root  required 


3.     15825(125  num. 
1 

22)056  ' 
44 


,245)1225 

1225 

That  is 


4. 
5. 
6. 


1 

1» 

2. 


=,755  Then,  ^/,  75=,  86602 
Jf—  ,  8706739526    Then,  v%8706739526=,93S09 
|-|—  ,  52439024  39     Then,  ^/,524S9024S9=,72414 

Square  root  of  mixed  numbers. 

Page  131. 
71^-=  is.!9     Then.  ^/1-^9=^?=6l 

'49  49  LUC11,    \  —    ?    -  U7 


Then,  ^^^=^=71- 
9|f/=\V     Then,  v/.W=¥=3-} 

Surds. 

%/7I\=v/75S18I81'&c.  =2,7961  + 

&c.  ^=2,9519+ 
&c.  =9,27+ 


94  SQUARE  ROOT. 

Any  two  sides  of  a  right  angled  triangle  given  to  find 

the  third  tide. 

Page  132.  ' 

1.     36X36=1296  2.     45x45=2025 

27X27=  729  60x60=3600 


Sum  2025  Sum  5625 

Now,  ^2025=45  feet  Then,  v/5625=75  yards 

3.     45X45=2025  AC  squared        4.    75x75=5625 
27X27=  729  AB  squared  45x45=2025 

Diff.  1296  EC  squared  Diff.  3600 

Then,  v/1296=36  feet  Then,  y/3600=60  feet 

To  find  the  side  of  a  square  equal  in  area  to  any  given. 

sup  erf  tecs. 

Page  133. 

1.     ^160=12,6491106  x/2025=*45 

3.     v/750=27,SS612r9 

To  find  the  diameter  of  a  circle  of  a  given  proportion,  larger 
or  less  than  a  given  one. 

1.  4X4=16,  and  16x4=64    Then,  <v/64=8  feet. 

2.  1002X3=SOOOO     Then,  v/300CO=173,20508  feet 

1^X12 

3.  -^-  —=72        Then,  N/72=8,4852814  inches 

% 

The  area  of  a  circle  given  to  find  the  diameter. 

RULE.     Divide  the  given  area  by  ,7854  and  the  quotient 
is  the  square  of  tlie  diameter. 


SQUARE   ROOT.  95 

i    Page  133. 


v/l60-7-,78D4=v/203,717850776674=14,272976 
Or,  by  the  rule  in  the  "Calculator." 
^160=12,6491106 
1,12837 


885437742 
379473318 
1011928848 
252982212 
1391402166 

14,272876927722  diameter 

2.  Because  160  perches  is  an  acre,  the  diameter  is  the 
same  as  in  the  foregoing  example.  Consequently  the  half 
diameter,  or  length  of  the  halter  will  be 

14,272976-i-2=7,136488  perches 
16£  feet 


114183808 
3568244 

feet  117,752052 
12 


inches  9,024624 
117  feet  9  inches  the  length  of  the  halter 

Application. 

Page  134. 

Qwtt.  1.     ^20736=144  men        2.    25x'25=625feet 
3.         N/1971 36=444  stones 

4.  600-^V=62V°=30  the  perpendicular.  Because  the 
perpendicular  and  the  other  side  is  the  same,  viz: — 30  per- 
ches, the  triangle  is  right  angled.  Consequently, 

^/402-j-303=v'1600+900=Sv/2500=x50  perches 


96  CUBE   ROOT. 

Page  134. 

5.  84x84=7056 

50X50=2500        Then,  v/9556=97,7547+mile3 

9556 

root  required          root  required 

6.  964,5192360241(31,05571      7.    1030892198,4001(32107,51 


61)  64  C2)  130 

61  124 

6205)35192  641)    f>S9 

310-25  _fi41 

62106)176736  64207)  482198 

372636  '  449449 

C21127)  4410002  612145;  :'>'J74940 

4347889  3210725 

6211341)   6211341  6481001)  64^1501 

6211341  6421501 

CUBE  ROOT. 

Page  138. 

•       • 

2.      2X2X3=12. .  13824(24  root  required 

square  of  4—     16  8 

2X4X30=  £40 

5824 

complete  divisor  1456x4=        5824 

3.    7X7X3-=147.  .        573248(72  root  required 
square  of  2=      04        343 

7X2X30=     420         

30248 

complete  divisor   15124X2=30248 

4.     1X1X3=3..  5735539(179  root  required 

square  of  7=  49          1 

1X7X30=210          

— — -          4735 
complete  divisor  559X7=3913 

17X17X3=867..  822339 

square  of  9=       81 
17X9X30=  4590 
complete  divis.  91371X9=822339 


CUBE    ROOT.  97 

Page  138 

5.       4x4x3=48 . .  84604519(439  root  required 

square  of  3=    09          64 
4X3X30—  360 

2C604 

complete  divisor  5169x3=15507 

4SX43X3—5547..  5097519 

square  of  9=        81 
43X9X30=  11610 


comp.  divisor  566391 X  9=5097519 

6.     3X3X3—27..  27054036008(3002  root 

defective  divisor  2700 . .  27 

defective  divisor  270000 . .          

square  of  2—  '           04  054036008 

306X2x30=       18000 


complete  divisor  27018004x2—54056008 

7.        4X4X3=48..  122615327232(4963  root 

square  of  9=     81  64 
4X9X30=1080 

58615       ' 

complete  divisor  5961  x9=        53649 

1080 

92X2=    162  4966327  dividend 

defective  divisor*  ^203 . . 

square  of  6—  36 

49X6X30=       8820 

complete  divisor  729156x6=     4374936  subtrahend 

8820  : * 

62  X2=  72  591391232  dividend 

defective  divisor  738048 . . 
square  of  8=  64 

496X8X30=        119040 


complete  divisor  73923904x8=591391232  subtrahend 

*  Defective  divisors,  after  the  first,  may  be  more  concisely  found  by 
addition,  thus:  To  the  last  complete  divisor,  add  the  number  which 
completed  it,  with  twice  the  square  of  the  last  figure  in  the  root;  the 
sum  will  be  the  next  defective  divisor. 

6 


98  CUBE    ROOT. 

Page  138. 

8.         2X2X3=  12..  2206981oi25(2805  root 

square  of  8=      64  8 

2X8X30=     480  — 

14069 

complete  divisor  1744x8=  13952 

480  

82x2=     128  117810125  dividend 

defective  divisor  235200.. 
square  of  5=  25 

280X5X30=         42000 


complete  divisor  23562025x5=117810125  subtrahend 

9.  6x6x3=108..  219365327791(6031 
defective  divisor     10800..         216 

square  of  3=  09          ~~S365327 

60X3X30=  5400 

complete  divisor  1085409x3=  3256227 

5400  

square  of  3x2= 18  109100791 

defective  divisor  1090827.. 
square  of  1=  01 

603X1X30=         18090 

|109:1 00791 XI =109100791 

root  rtquind 

10.  8X8X3=192..  673373097125(8765 
square  of  7=       49  512 

8x7X30=  1680  T61373 

complete  divisor  209*9x7=       146503 

1680  14870097 

twice  the  sq.  of  7= 98 

defective  divisor  2^707.. 
square  of  6=  36 

87X6X30=     15660 
complete  divisor  2286396x6=    13718376 

15660  1151721125 

twice  the  sq.  of  6= 72 

defective  divisor  2302128., 
square  of  5=  £5 

876X5X30=       131400 

complete  divisor  230344225X5==  115172 11 25 


CUBE   ROOT. 


99 


Page  138. 


11.     2X2X3=12.. 
square  of  3=     09 
2X3X30=  180 
complete  divisor  1389x3= 

180 

twice  the  sq.  of  3=     18 
defective  divisor  1587. . 
square  of  5=  25 

23X5X30=       3450 
complete  divisor  16£175x5=810875 


12,9778f5(2,35  root 
8 

4977 
4167 

810875 


12.      £X2X3= 

square  of  5= 

2X5X30= 


25 
300 


complete  divisor  1525  X5* 

300 

twice  the  sq.  of  5=     50 
defective  divisor  1875  .  . 
square  of  1  =         01 
25X1X30=        750 

complete  divisor  188251x1  = 

750 

twice  the  sq.  of  1=  _  2 
defective  divisor  189003.. 
square  of  6=  36 

251X6X300=        45180 
complete  divisor  18945516x6= 


15926,972504(25,16+ 
8 

7926 
7625 


301972 


188251 
113721504 


•113673096 


48408  rem. 


36155,027576(33,06 

27 

9155 

8937 


13.     3x3x3=  27.. 
square  of  3=      09 
3X3X30=     270 
complete  divisor  2979x3= 
33X33X5=3267..  "21802/576 

defective  divisor  326700 . . 
square  of  6=  36 

330X6X30=       59400    ;/ 
complete  divisor  32729436X.6  =196376616 

21650960  rem. 


100  CUBE   ROOT. 

Page  139. 

14.     5x3x3=  27..  ,0532582/9(,S76-froot 

square  of  7=      49  27 

3X7X30=     630  — 

£6258 

complete  divisor  3379x7=    23653 

37X37X3=4107..  2605279 

square  of  6=         36 
37X3X30=     3330 


complete  divisor  414066x6=  2484396  ' 

120883  remainder 

15.     1  xl  X3=3 -. .          ,001906624(,124  root 
square  of  2=    04  1 

1X2X30—    60 

906 

complete  divisor  364x2=       728 

12X12X3=432..  178624 

square  of  4=      16 
12X4X30=  1440 

complete  divisor  44656x4=178624 

16.  ,OOQ000729(,Q09  root  required 

729 

17.  1X1X3=3.,  2,(l,25+roet 
square  of  &=    04  1 

1X2X30=     60  

1000 

complete  divisor  364x2=    728 

12X12X3=432..  JJOO 

square  '•• 

1X30=  1800 


complete  divisor  40035x5*=    225125 


\46875  remainder 


CUBE   ROOT.,  101 

Page  139. 

To  extract  the  cube  root  tf&wlgw  'frty&fa 
1.     If  «=4ff    Then,  ^|f  Hr  root  required 

Then>  ^AV=I  root  required 
Then>  ^2T=I  root  required 

Surds. 
4<     ^4  =  ^571428571  &c.  =,829+root  required 

5.  ^/|=^/5666  &c.  =5873+root  required' 

6.  ^/|=^/5555  &c.  =,822+root  required 
To  extract  the  cube  root  of  a  mixed  number. 


3. 

Surds. 

Page  140. 

4.  -\^7y=  ^7,2=1,93-}-  root  required 

5.  ^/8f  «=  ^8,7142857  &c.  =2,057+root  required 

6.  ^/9|=:^/9,166  &c.  =2?092+root  required 

To  find  the  side  of  a  cube  that  shall  be  equal  to  any  given 
solid,  as  a  globe,  a  cone,  tyc. 

1,     ^10648=22  2.     ^389017=73 

Note. 


Then,  -^24  =2,8  845  feet  =2  feet  10,614  inches 

.     123XS=12X12X12X3=5184 
Then,  ^5184=17,306  inches 


ARITHMETICAL    PROGRESSION 
Page  140. 


1.     As  6-    :  33  v  216  :  27  ::  32  Ib.    :  4  Ib. 
3.     288x216x48=2935984     Then,  ^72985984=  144 
3,     i!/3S9017=73     Then,  73X73=5329 

Page  141, 

4.     Because  there  is  two  half  feet  in  a  foot 
--1-^  —  =|=4  solid  half  feet  in  half  a  solid  foot 

And  then,  4—1=3  solid  half  feet 

5.  2X2X2=8  sis  inch  cubes.  And  because  there 
are  three  times  4  inches  in  a  (£ot  we  find  that  there  is 
3x3x3=27  four  inch  cubes  in  a  solid  foot 

ARITHMETICAL  PROGRESSION. 

Case  1.     Page  142. 

2.  (20—1  X3)+3=(19X3)+3=60  last  term 
Then,  (60+S)x  Y=63Xl£r=6  dollars  30  cents 

3.  Here  4  is  the  common  difference 

And  (100—  1X4)+4=(99X4)  +4=400  last  term 
Thea.   (400  +  4)  X1!0  =404X50=20200  yards 
But,  20200  yards=ll  ir/iles  3  furlongs  180  yards 

Page   143. 

4.  (ToITixlO}  +  20=90--f-20=110  dollars  last  term 
Then,  (110-j-20)X  ^=130x5=:  650  dollars  the  \O 

amount.     And  650-7-10—55  dollars  per  annum 

~~oITix  10;  -f  10~999G-rlG—  10000   cents,    or 
100  dollars  for  the  last  acre 
And  (100CO-f-10)x10s00=50050  dollars,  whole  cost 


GEOMETRICAL   PROGRESSION.  105 

Case  2.    Page  143. 

2.  — •=|$=2  the  common  difference 
Then,  20,22,24,26,28,  &c.  are  their  ages  respectiyely 

Page  144. 
^g 3 

3.        -— ff=5  miles  daily  increase 

And  (58+3)  XV2— 61x6=366  miles  whole  distance 
GEOMETRICAL  PROGRESSION. 

Page  146. 

2.        Indices     1    2    S    4 

Ratio       2,  4,  8,  1 6,  powers  of  the  ratio 

^6 

In  this  question  the  256:=8th  power 

number  of  terms  is  1£  S— 3d  power 

2048=1 1th  power  of  the  ratio 
X  1  the  first  term 

[progression 

2048=:  12th  or  last  term  of  the 
X  2  the  ratio 

4096 
Subtract        1  the  first  term 


Ratio  2—l=l)409o 

84095  the  sum  of  the  series 

3.  Indices    12345 

Ratio    2,  4,  8,  15,  32,  powers  of  the  ratio 


(54 

Here  the  number  of         98 
terms  i*  15 

1024=1  Oth  power 

16=4th  power 

I(j;i84=14lh  power  of  the  ratio. 

And  because  the  first  term  oi'  the  progression  is  1,  the  14th  power 
of  the  ratio  is  the  ia^t,  or  15th  term  of  the  progression 


TheBj  Z.32767  Aillinp=163&e.  T*. 


104  GEOMETRICAL    PROGRESSION. 

Page  146. 

4.  Indices     1234 

Ratio       4,  16,  64  256  powers  of  the  ratio 
256 

1536 

Here  the  number  of      1280 
terms  is  12  512 


655S6=:Sth  power 
64=3d  power 


262144 
393216 

4194304— llth  power  of  the  ratic 
which  is  also  the  12th  or  last  term  of  the  progression. 

Then,  (4194^Xlhz^55924  05  sold  for 

And  12X20—240  00  cost 
855684  05   gain 

5.     Indices  123456       7 

Ratio  2,   4,    8,  16,  32,  64,  128,  powers  of  the  rati« 

128 

Here  the  number  of  1024 

terms  is  32  256 

128- 


16384=14th  power 
26384=.      do. 

65536 
131072 
49152 
98304 
16384 

268435456— 28th  power 
8~3d  power 


2147483648=3lst      power     of 

the  ratio  i  And  because  the  first  term  of  the  progression  is 
1,  it  is  also  the  32d  or  last  term  of  the  progression 
Then,  (2^-483648  X2)-l=4a949  29         5  mg 

— 


2  —  1  Ratio 


GEOMETRICAL   PROGRESSION,  1.05 

Page  148. 

6.     Indices  12345 

Ratio     3,  9, 27, 81, 243  powers  of  the  ratio 

243 

729 
972 
486 

59049=10th  power 
59049=       do 
531441 
236196 
5314410 
295245 

34867B4401=20th  power 
59049=10th  power 


31381059609 
13947137604 
313810596090 
17433922005 

205891  132094649=30th  power 
3=single  power 

617673396283947=*31st  power  of  the  ratio,  or 
S2d  term  of  the  progression 


Then,  --=926510Q94425 

Ratio    3  —  1 


'  (432—l).  1844674^073709551615 


3 

6148914691236517  dollars  20  cents  5  mills 

(330~- 1)     2X205891132094648 
___  __ 

205891132094648  pins    Which  at  1000  for  a  farthing 
amounts  to  £. 214469929  5  S£ 

cost  of  30  yards  at  ^100^  3000  0  0 

gain  £.214466929  5  3£ 


SINGLE    POSITION. 

SINGLE  POSITION. 
Page  148. 

S.        Suppose  A  paid  12  dollars 
Then      B  4      < 

And       C  1      < 

17  sum 

dolls.        T12  :  240  A  paid 
Now,  as  17  :  340  ::     <    4  :    80  B     < 

i  1  :  20  e     « 

4.  Suppose  60 

{IS  Now,  as  37:  60::  148:  ^==240  d. 
J  =  10 

S7 

5.  Suppese  12 

£=  6 

1=  4  Then,  as  2  :  12  : :  26f  dolls.  :  160  dolls. 

10 
left  2 

6.  Suppqse  A's  age    6 
Then,B's=6xl|=9 

AndC's=(6+9)x21V=31| 


*tu?^ 

r 6  : 

6,5  :  95  ::     -{9     : 
C31,5: 


.    -     .  12  A's 
Then,  as  46,5  :  9$  ::     ^    9     :  18  B's 

63  C's    « 

7.    Suppose  1  to  be  the  number 
Then,  I— 1=°  17°  -=A  difference 


DOUBLE   POSITION.  107 

Page  148. 

8.    Suppose  1  the  harness  -3EPH  B  HAMMOJsl) 

then        £  '    horse 
and         6   *    chaise 


harness 

horse 

chaise 


y 

dolls,         f  1  :  40  dollars  harnei 

Then,  as  9  :  360   ; :    1  2  :  80  dollars  horse 

1 6  :  240  dollars  chaise 

9.     Suppose        60  10.    Suppose  12 

5=-  4 

1=20  — 

1=15  A  spends  8 

|=12  2 

£=»10  — 

B  spends  16 

57  12 

B  sinks    4 

dolls,  dolls.  dolls,  dolls. 

Now,  as  57  :  60  ::  228  :  240  Then,  as  4  :  12  ::  50  :  150 

DOUBLE  POSITION. 

Page  150. 
2.     Suppose  8  Again,  suppose  40 

A  spends    7  A  spends  35  per  year 

30  30 

B  spends  37  per  year  B  spends    65  per  year 

8  8 

B  spends  296  in  8  years          B  spends  520  in  8  yrs. 
8X8=*  64  40X8=320 

B  is  indebted  232  B  is  indebted  200 

40  40 

192  error  too  little  160  e.  too  lit, 

192  error  X40=7680 
!60errorX  8=1280 


6400  difference  of  the  products 
192—160=32  the  difference  of  the  errors 


108  DOUBLE    POSITION. 

Page  150. 

Question  2d  continued. 

Then,  6400—32=200  dollars  their  income. 
And  200— -2£°  =200— 25=175  dolls.  A  spends  per  annum 
Also,  175+30=205  dollars  B  spends  per  annum 

S.    Suppose  A's  12         Again,  suppose  A's  40 
Then  B's  is  15  Then  B's  is  43 

And  C's    *    19  And  C's  «  47 

46  130 

100  100 

Error    54  too  little  Error  SO  too  much 

54X40=2160 
30X12=  360 


2520  sum  of  the  products 
And  54 + 30=84  sum  of  the  errors 
Then,  2520-i-S4=30  A's  share 

30+  3=33  B's  share 

33+  4=37  C's  share 

4.    Suppose  A  paid    100      Again,  suppose  A  paid     200 
then  B      4     1100  then  B     •      1200 

and    C       *     1200  and  C     '      1400 

2400  2800 

10000  10000 


error  7600  error    7200 

Errors  are  alike,  i.  e.  both  too  little 
7600X200=1520000 
7200X100=  720000 


Diff.  of  the  products,  800000 

7600—7200=400  difference  of  the  errors 
Then,  800000—400=2000  dollars  A  paid 
Aad  2000+1000=3000  dollars  B  paid 
Al69_2000+3000=5000  dollars  C  paid 


DOUBLE    POSITION.  109 

Page  150. 

5,  Suppose  first  horse  §10    Again,  sup.  first  horse  820 
50  50 

2)60  2)70 

The  second  horse  30         Then,  second  horse  35 
50  50 


80 
10X3=30 

85 
20X3=60 

Error  too  little    50 

error  too  little  25 

50X20=1000 
25x10=  250 

50 

25 

Diff,  of  the  prod.    750      DifF.  of  the  errors  25 
Then,  750—25=30  dollars  first  horse 


& 


^o  dollars  second  horse 


6.     Suppose  20  body      Again,  suppose  50  bod j 

Then,   2/+9=19tail         Then,  3^° +9=24  tail 
9  head  9  head 

Head  and  tail =28  Head  and  tail  33 

body=20  body  SO 

error  8  too  little  error  3  too  little 

50x8=240        8 — 3=5  difference  of  the  errors 
20x3=  60 

Then,  180—5=36  inches  the  body 

Diff.  of  pro.  180        And^-f9=27  «    tail 

9  *    head 

72  in.=6  feet 

10 


110  DOUBLE    POSITION. 

Page  150. 

7.  Suppose  he  wrought  £0  days,  and  was  idle  £0 

Then,  20  days  X  20  cents =400 

20     '     xlO     «     =200 

He  would  receive  200  cents 
500 

error  too  little  300 

Again,  suppose  he  wrought  25  days,  and  wag  idle  15 

Then,  25  daysXSO  cents=500 
15     k"    xlO      <     =150 

He  would  receive  350 
500 

error  150  too  little 
25X300=7500 
20x150  =  3000      300 — 150=150  difference  of  tke  error* 

Diff.  of  pro.  4500 

Then,  4500—150=30  days,  wrought 
And  --10  days,  idle 

8.     Suppose  each  had  300  dollars 

;}-J-3!°=300  +  75=375 
And  5)X2=75X2=150 

Difference  225  error 
ppose  each  had  400  dollars 

IVn.  400  +  4J0=400-H°0=500 
And  (400— i25)x2=rl75x2=35Q 

Difference  150  err . 

like 

Now,   4 OCX 225 =90000 
X  150=45000 

DifF.  of  the  products  45000 

An<]  :3J — 150 -=75  difference  of  the  errors 
Then,  45GO-r-75=600  dollars 


PERMUTATION.  HI 

Page  150. 

9.     Suppose  the  parts        Again,  suppose  the  parts 
to  be  8  and  7  to  be  10  and  5 

Then,  Sx  4=  32  Then,  10  X  4=40 

7X16=112  5x16=80 

80  error  40  error 

Errors  alik«. 

10X80=800  80 — 40=40  difference  of  the  errors 

8X40=320    Then,     480—40=12  the  greater 

DIff.ofpro.  480  And  15—12=3  the  less 

Page    151. 
16.     Suppose  1  o'clock 

Then,  (T23-  of  ll)c«  1==^13  ^  ^rror 

i  \j 
Agaia,  suppose  it  to  be  two  of  the  clock/5 

0(5 OQ 

Then,  (ft  of  10)c*s  2=- — -^  =T*^  error 

1  o 
Errors  unlike 

Now,  T'jX2=if,  and  &xl«& 
Then,  •}§•+•& =fj  ^e  sum  °^  tfc 
And  TT+l*?3*!!  ^e  sum  °^  t^e  errors 
Lastly,  ^^1=11x11=14=1  hour  36  minu 

PERMUTATION. 

2.  changes 

1x2X3X4x5x6x7x8x9x10x11x12=479001600 

10)479001600  changes 
daythrs.      ^)J7900160 
365   6=8766)       798336(91  years 
78894 

9396 
8766 

days  w.  d. 

24)   630(26=3  5 
48 

144 

6  hours. 
Ans.  91  years  3  weeks  5  days  6  hours 


1  IS  COMBINATION. DUODECIMALS* 

Page  151. 

5.     1X2X3X4X5X6X7=5040  days. 
Then,  5040-7-365=13  years  295  days 

Page  152. 
4.     1X2X3X4X5X6X7X8X9=362880 

5.     1X2X3X4X5X6X7X8X9X10X1 1X12X13X14X15 
X16X17X 18X19X20X21X22X23X24X25X26= 
403291461 126605635584000000 

COMBINATION. 
4  11    7     £ 

0^X23X00X^X00X19 


5  '  £  13     5 

5X29X27X13X5X23=5852925 

ADDITION  OF  DUODECIMALS, 

Page  153. 


m.  w  "'  '"' 
2,    S780  9479 

SUBTRACTION  OF  DUODECIMALS, 

Page  154. 

2.     916  feet  8  inches  1  second  8  thirds  2  fourths 

//.  in.  "  '"  ft.  in.  "   '"  "" 

S-       From  35    920  From  803  3  4     00 

Take  24  10    55  Take    70  3  7  10    0 


ifi:   10  10   87  Diff.  732  11  8     1    S 


DUODECIMALS. 


113 


MULTIPLICATION  OF  DUODECIMALS. 

Case  1.    Page  155. 


ft.  in. 

//.  in. 

ft.  to." 

2. 

9  5 

3.       7  10 

4.       846 

3  11 

8  11 

2 

7  4 

877 

722 

2  9 

6 

28  3 

62     8 

4  10  7 

6 

1fi     Q  O 

square  ft 

36  10  7 

sq.  ft.  69  10    2 

A  VJ       y    \J 

square  feet  21  10  5 

0 

feet  in. 

5, 

5    7 

Then,     150  cents 

I   10 

10  square 

feet 

4     7  10 

2  in.     -( 

S-       1500 

5     7 

8   "      - 

25 



2     "           2 

\            81 

square  ft. 

i 

10     2  10 

3& 

ft.  in. 
7  10 
6  8 
5  4 

19  10 
3  11 

18     2     2 
59    6 

77     8     2 
3 


Then,  233  square  feet 
14  cents 


3262 

- A 

832,62^ 


0    6  square  feet 
10* 


114 


DUODECIMALS 


Sf 

Case  2.     Page  156. 

2, 

ft. 

76 

in. 

7X1 

.      //.    in. 

S.       127    6x2 

//.    in. 
4.     184    8X7 

459 

6 
6 

10 

12 

1275  0 

2216  0 

3 
6 

9 

10 

1378 

11475  0 

22160  0 

in. 

76 

7 

in.          255  0 

in.          1292  8 

6  is 
4  ; 

|=  38 

3     6 
6    4 

4is|=  42  6 

6is£=     924 

11772  6       i 

?n.  fret  23545  0 

sq/ft.  1518  10  10 


Page  157. 


6. 


6  in.  is 


feet  in. 

38  11 

10 

389     2 

7 

2724     2 
19    5     6 


6  in.  is 


square  feet  2750     1    4 


1434     0 

=    £9  10| 

9)1463  Wk 

162 yds.  5 ft.  10|in, 


8. 


fest  in.       feet 
54  9=54,75 
58  6=  58,5 

27375 
43SOO 

27375 

9)3202,875 


355,875  yards 
15  ceiita 


1  10  6 

22     6 


20     3  9     0 
24     4  6 


853,33125 


square  feet  44    8  3    0 


Page  157. 

ft.  in,    ft. in.  ft.  in.    ft.  in.  " 

10,           (2  10)X(2  4)x(l  9)  =  11  6  10 

(2  10)X(2  6)X(1  3)=  8  10  3 

(3     6,X(2  2)x(l  8)  =  12  7  B 

(2  10) X (2  8)X(1   9)  =  1S  2  8 

(2  10)X(2  6)x(l  9)  =  12  4  9  "' 

(2  lb)x(2  8)X(1  8)=12  11  6  8 


71     7    8  8 
Or,  71,6435  square  feet 

ft.  dolls.  sq.  feet. 

Then,  as  40  :  £0  v  2  :  1  :  :  71,6435  :  835,82+ 

To  find  a  ship'*  tonnage. 
Page  J58. 


3, 


4. 


THE  CARPENTER®  OR  SLIDING  RULE. 

Page  160. 

1st.     To  multiply  numbers  together. 

2.  Set  1  on  B  to  12  on  A,  then  against  16  en  B  stands 
192  the  product  on  A. 

3.  Set  1  on  B  to  35  on  A,  then  against  19  on  B  stands 
665  the  product  on  A. 

4.  Set  1  on  B  to  54  on  A,  then  against  270  on  B  stands 
14580  the  product  on  A. 

2d.    Division  of  numbers  by  the  Carpenters*  rule. 

£.  Set  the  divisor  19  on  B  to  the  dividend  665  on  A, 
then  against  1  on  B  stands  35  the  quotient  on  A. 

3.  Set  27  on  B  to  396  on  A,  then  against  1  on  B  stands 
14,6  the  quotient  on  A. 


116 

Page  160. 

4.  Set  42  on  B  to  741  on  A,  then  against  1  on  B  stands 
17,6  the  quotient  on  A. 

5.  Set  24  on  B  to  7680  on  A,  then  against  1  on  B  stands 
320  the  quotient  on  A. 

Sd.     To  square,  numbers  by  the  Carpenter?  rule. 
Page  161. 

2.  Having  set  1  on  C  to  10  on  D,  then  against  30  on  D 
stands  900  on  C. 

3.  Having  the  rule  set  as  above,  against  55  on  D  jou 
will  find  1225  its  square  on  C. 

4.  Also  against  40  on  D  stands  1600  its  square  on  C. 

4th.     To  find  a  fourth  proportional  to  three  numbers. 

2.  Set  the  first  term  25  on  B  to  75  the  second  term  on 
A,  then  against  100  the  third  term  on  B  stands  300  the 
fourth  term  on  A. 

3.  Set  27  on  B  to  20  on  A,  then  against  73  on  B  stands 
54     on  A. 


5th,      To  extract  the  square  root  of  any  number  by  the 
Carpenters'  rule. 

2.  Having  set  1  on  C  to  10  on  D,  against  529  on  C 
stands  23  its  root  on  D. 

3.  Being  set  as  in  the  last  example,  against  900  on  C 
stands  30  the  root  on  D. 

4.  The  rule  set  as  before,  against  300  on  C  stands  17,3 
the  root  on  D, 


MEASURING   OF   BOARDS,    ETC.  117 

MEASURING  OF  BOARDS  AND  TIMBER. 

Page  162. 
1st   To  find  the  superficial  content  of  a  board  or  plank. 

foot  in. 
2.        12  Or,  by  the  sliding  rule. 

12    6  As  12  on  B  :  14  on  A:: 

12  ft.  6  in.  on  B  :  14  ft. 

7  in.  on  A*. 


==70l=2f 3  inches 
feet    V  feet 


4x12 

And  171  feet=325  feet 
Then,  \Vxyx3  cts.=^~^=9||5=r3d.  09f£  cti. 

2d.    Having  the  breadth  of  a  board  or,  plank  in  inches  j 
to  find  how  much  in  length  will  make  afoot,  fye. 

Page  163. 

5.     144-r-23=6,26  inches 
S.     1|  yards=1944  square  inches 
Then,  1944-7-26=74,7692  inches=6,2307  feet 

3d.  To  find  the  solid  content  of  square  or  four  sided  timber. 
Page  164. 

15  in. Xl5in.xl8  ft. 
144~~ 

25  in.  X9in. 
3-     5" **tf  1!lches,  the  quarter  girt 

17in.xl7in.x20ft. 
Then,  ^ =40,1388  feet,  &c. 

*The  operation  by  the  sliding  rule  is  omitted  after  this,  it  beinr 
presumed  that  the  direclions  in  the  W.  Calculator  make  it  superflu- 
ous to  work  more  examples. 


116  cAmpfwraRS5  AND  JOINERS'  WQKK. 

Pa&e  164. 

SSin.-flOin. 

-  2  -  tea     mc«es  the  mean  breadth 

And  -  ^-  -  -=*18  inches  the  mean  bread  th 


8 

4th.     To  find  the  solid  content  of  round  timbtr. 

Page  165. 
14X2 


2X4 
Then,  2x2x24=96  feet. 

Page  166. 

5.     24  inchea=2  feet 
Then,  £  ftetXS  fcetx  IJB  feet«=7S  feet. 

4.  nft4in.+2ft.8in. 

2X4 
Then,  1,75  ft.  X  1,75  ft.  X21  ft.=64,S125  feet 

5.  24  x  (|)2  =24X4  =96 
14ix(Y5)2  =  l4,5x,  62015625-=  8.9922+ 
17!x(6;T)2=17,25x2,4649     =42,5195 

sum  147,51  17  feet 

CARPENTERS*  AND  JOINERS'  WORK. 
Page  168. 

fc    (5Sft.6in.)x(47ft.9in.)     53,5x47,75     ,?!:**. 
2L  —  =-  =—  >^o  o4«0>oO 

100  100 

,    (36ft  Sin.)  x(16  ft.  6  in.)     36,25X16,5 

§,  —  ---  .  =  -  s 

100  100 

Thtti,  5?98125  8quaresx4  d.  50  c.=26  d.  91  c.  5,625  1 


CARPENTERS.'   AM)   JOSSERS*    WORK. 

168'. 


4.        35  ft.  X30  ft.  =1050  square  feet 

(4ft.  6in.)x6ft.=  4,5x6=27  square  ft.  fire  place 
(I0ft.6in.)x8ft.=10,5x8=84      «        <  stair* 

111  feet 

Then,  1050""m=9,39  squares 
100  , 

Lastly,  9,39  squares  X  3d.  75c.=35d.  21  c.  2,3  mr. 

/?.  in.    ft.  in.     ft.     ft.     sq.   feet 
5    (82  6)x(12  3)     8a,5Xlg4  =  1Q  1Q  6gg 


100  100 

(91ft  9in.)x(llft.  3  in.)     91,75x111 

"~ioo~  "Too^ 

squares       dolls,  cts*      d,     ctf.      milli 

Then,  10,321875x4  50=46  44  8,4375 


ft.  in. 
8,     2)30  6 


Page  169. 
15  3 


45  9~45,75  feet.    And  52  feet  8  in.=52f  feet 
Then,  1^   pf  «24,095  squares 
Lastly,  24,095  sq.xHO  cts.=S3  dolls.  7S  cts.  5  raille 

feet  in. 
9.     20  6 
i  =  lO  3 


30  9=30,75  feet.    And  40  feet  6  incfees=4Q,5  feet 
Then,  ?0'7*^°'5»lg,45375  squares 
Lustly,  12,45375  squaresx225  cts.  =28  dollars  2  centa-f 

Page  170. 

iquare  ftet 
11.  (137  ft.  6  in.)x(16  ft.  3  in.) =137,5x164 =2234,375 

Then,  2234,375  feet-^9=248  yards  2,375  fe«t 


1£0  MASONS9    WORK* 

Page  170. 

!£.      (69  ft.  9  in.)x(6  ft.  3  in.) =69, 75x6^=435,9375 

I  work=217,96875 


9)653,90625 
square  yards    72,65625 

13.    (83ft.  8in.)x(12ft.  6  in.)=83fx  12,5=1045,831 
(7ft.  8in.)x(2ft.  6in.)xli=7fx2,5xl,5=r=     £8,75  wind. 
(3ft.  6in.)X  7ftx£  =3,5x7X,5  =     12,25door 

9)1086,831 

square  yards     12057|f 
Then,  120,7if  Jds-  x80  cts.=96  dolls.  60  cts.  7\\  mills 


ft. 

(57 


BRICKLAYERS'   WORK. 

Page  173. 

t.  in.    ft.  im.  half  brick 

(57  S)X(24  6)X5~57,25X24,5X5=7013,125 

Then,  3)7013,125 

9)2337,708^  standard  thickness 
25  9,74  5  -f-  square  yards 
Or,  2337,7081—272^=8,5866  rods 

feet  in.     feet  in. 
5.     (245  9)X(16  6)X5=245,75X16,5X5=?20274,575 

And  20274.S75  -r-3=»6758,125  standard  thickness 

Then,  6758?125=24  rods  3  quarters  19?945  feet 

272,25 

4.     (45  ft.  x  V  ft-  x5)-f-3=562,5  standard  thickness 
Then,  562,5-^-272,25=2  rods  18  feet 

MASONS?  WORK. 

Page  175. 
feet  in.  feet  in. 
*.     (53  6)X(12  3)x2==53,5Xl2|x2=lS10,75fcet 

Then,  1310,75—24,75=52,9595  rods 


PLASTERERS'    WORK. 


Page  175, 
(  107  ft.  9  in.;  x  (20  ft.  6inJ=*2208  feet  10J  inches 

(112ft.  S  in  J  x£16  ft.  6inJ     1  1  2,  25  X  1  6,  5     «"*•    /««  _ 
~63~  ~~6™ 

5.     fa  ft.  7  in.;x(l  ft.  10  in.;=f.|x||=  T^V^VV  feet 

7^7x  1  0 
Then,  V'/  X30  cts,=:^~  =-318-1  cts.  =8  dolls.  18|  cts. 


PLASTERERS'*   WORK. 

Page'  177. 

0      (141  6;x(U   3)_141.5X11,25_^^ 
9  9""  '  ' 

3.     ^22  ft.  7injxf  13  ft.  11  iuJ=V/  x¥/=4i-!i7*  feet 
But,  <«*'  ^et= 


Then,  4T522A7X15  cts.=22^3V5=5doils-  23  cts. 

4.  (30  ft+14  ft.  6  in.j  X2==     60  feet  round  the  room 
69  ft.X(10  ft.  4  in.j  =713  sq.  feet  in  the  walls 

(4  ft.  4  in)  x4  ft  =     17  ft.  4  inches  fire  place 
(S  ft.  2  in.)  X6  ft.  X2i=     38  ft  0     ,         two  windows 

Sum  55  ft.  4in.=55^  feet 
713  ft—  55l=657f  Lastly,  657|-7-9^73/f 

5.  (14ft  5  in.+lS  ft.  2in.)x2=55  ft  2  inches  round 

ft.  in.  " 

Then,  (55ft  2  in.  )X  (9  ft.  3  in)=510    3   6 
7ftx4ft=  28    0   0 


9)482    3    6 

Rendering  53  j.  5  ft.  3  in.  6  s. 
ft.  in.          in.     ft.  in.  ^ 

14  5  less  10=13  7  And  15  ft.  2  in. —10  in.=  12  ft.  4  in. 

TheD,  (!iM 
11 


22  PAVERS'  WOHK. 

Page  177. 

feet    in.       feet    tn. 

(105  6)X(275  5)_105,5X275T 

9  feet  9 

Then,  S22S,4}-££-  sq.  yds.  X12  cts.  =  387  d.  41  c.  9|ra 

/ee£  tn.     feet 
7*    (30  ft.  6  in.-f  24ft.  9  in.)x2=110  6  =  110,5  round 

110,5  ft.  XlO  ft.=  1105  square  feet  in  the  walla 
30,5  «  X24J  <  =   754,875  in  the  ceiling 

9)  1859,875 

206,652|  square  yards 
2  cents 


Or,  4  dollars  13  cents  3  mills. 
PAVERS'  WORK. 

Page  178. 

(35. ft.  4in)x(8  ft  3in.)_35^x8,£5 

9  9 

Then,  32,3|-  sq.  yds.x28  cts. =9  dolls.  06  cts.  8|  mills 

5.    (27ft.  10in.)x(14ft.  9in.)=27JXl4,75=410,54J  g.f. 
410,54-J— ^9=45,6l|^  square  yards 

Then,  45,61^4x38  cts.  =  1733,S9|f  cts.=  17  S3  S,9-f 
4.    45  feet — 5  feet  3  inches=S9  feet  9  inches 

broad 


Then,  56|  square  vardsXS6  cts.  =  13  23 
278i         <     "   *     X30     '    =83  47i 


Sum  896  70i 


123 
PAINTERS'  WORK. 

Page  179. 

ft.  in.  " 
2.     (74ft.  10 in.) X (11  ft. Tin.)  =     866  9  10 

(7ft  6in.)x(3  ft.   9  in.)=      28  I     6  door 
(6ft  8  in.)x(3ft  4in.)x5=     111  1     4  windows 
(22  ft.  8in.)x(l  ft.  2  in.)x5  =     132  2     8  breaks 

1138  3     4 
(6  ft.  9  in.)  X5  feet=     3390  chimney 

feet 

1104  6    4=1104f| 

And  1104i£-7-9=122ff  £  square  yards 
Then,  122fff  s.  yds.x8<|  cts.=10  dolls.  43  cts.  If  J£  ms. 

5.    (20  ft+14ft.  6  in.)x2=69  feet  round 

69  ft  X(10  ft.  4  in.) =69x10^=713  square  feet 

(4  ft  4  in.)x4ft=~4^x4=       17|  feet  fire, place 
(3ft.  2in.)x6ftx2=3£x6x2=38  feet  windows 

Sum  55| 

5£     6574 

*  =73^2T  square  yards.* 

Page  180. 

ft.      in.  " 
4.     (24ft.  6in.Xl6ft.  3in.)x2=1039     16 

(3  ft.  6  in.)  X7  feet  =     2460  door 
(7ft  9in.)x(Sft  6tn.)x2=     54     3  0  w.  shut. 
24  ft  X(l  ft  3  in.)  X2=     60    0  0  breaks 

1177  10  6 
(5ft  6in.)x5feet=     27    6  0 


1150    4  6 

9)iiaoj 

127« 

Then,  127|Jx6  cts.  =7  dolls.  66  cts.  9-J- ms, 
*Tbe  same  as  question  4,  plasterers'  work. 


124  MEASUREMENT    OF    GROUND, 

GLAZIER®  WORK 

Page  181. 

2.  (7ft.  5  in.)x(3  ft.  5  in.)  =24  square  feet  9  in.  3  secJ 

3.  (6  ft.  1  1  in.  +5  ft.  4  in.-f  4  ft.  3  in.)  X(3  ft.  6  in.)  X7= 
16,5X3,5X7=404,25  square  feet 

Then,  404,25  sq.  ft,  Xl4£  cts.  =58  d.  61  cts.  6,25  ras. 

4. 


Then,  104,6875  sq.  ft.  xlO  cts.  =10  d.  46  cts.  8|  ms. 

5.  (14  ft.  6  in.)x(4  ft.  9  in.)=68  sq.  ft.  10  in.  6  sec. 

6.  (7ft  10in.+6ft.  8in.+5ft  4in.)x(3ft.  Ilin.)x9= 


19fx3l|-x9=-^XT^X0=5V3=699|  square  feet 

£>      $#1 
2       4 

Then,  699|  square  feetx!4  cents=97  dollars  87*  cents 
MEASUREMENT  OF  GROUND. 

1st.     To  find  the  content  of  a  square  piece  of  ground. 

Page  162. 

35x2=70  perches 
Then,  70x70=4900  square  feet=30  A.  2  R.  20  P. 

sq.  perches     Jl.  R*    P- 

3.      16|  x  16|  ;  =  16,5  X  16,  5  =-£72,  25==  I  2  32i 


2d.     To  find  the  content  of  an  oblong  piece  of  ground, 
called  a  parallelogram. 

Page  183. 

120  perches  X  84  perches 
2:  —=63  acres 

perches 


MEASUREMENT    OF    GROUND.  125 

Page  183. 

!40^0;=58SOO=1()5 
16£X16£      272i 
But,  105  per.  213|=2  rods  25  perches  213|  feet 

3d.    To  find  the  content  of  a  triangular  piece  of  ground. 
Page  184. 

2.  75 XV  =75X25=1875  per.=ll  A.  2  R.  35  P. 

3.  120X8/«=120x42=5040  per.=31  acres  2  roods 

A.  R.  dolls.  dolls,  cts. 

Then,  (31  2)x45=31,5x45=1417  50 

4.  140  ft.x72°ft.=140X35=4900  square  feet 
Then  4900-7-9=544  yards  4  feet. 

4th.    To  find  the  content  of  apiece  of  ground,  in  the  form 
of  an  oblique  parallelogram. 

Page  185. 
2.    (80X24)~160=12  acres 

5th.    To  find  the  content  of  a  piece  of  ground  bounded  by 
four  sides,  none  of  which  are  parallel  or  equal. 

Page  186. 

120  X  (48+24) 
2.    ~- -=4320  perches=27  acres 

6th.     To  find  the  area  of  a  piece  of  ground  lying  in  a 
circle,  or  ellipsis. 

Page  187. 

perches    A.  R.  P. 

2.  30X20X, 7854=471, 24=2    3  31,24 

3.  160xl60x,7854=20106,24  square  feet 
But,  20106,24-7-9=2234,02|  square  feet. 

11* 


MECHANICAL    POWERS. 

GAUGING. 

Page  169. 

2.     £3+f(24—  18)  =  18+4==22  mean  diameter 
4 


Then,  «^|69||  gallons 

-' 

3.     oS-L|(36—  28)=28-f5;;=S3f  mean  diameter 

20 

_  .  X  S  S  •!  X  40      1  00  X  1  00  X 

TnPii      ___  __  —  200000 

.ineB^            294                9x  "  TT'2"? 

147 

4.     154-|(18—  15)=15  +  2=17  mean  diameter 

\~  \t  17  x2Q 
TheC'     '    359       ^  ^V  =23*^  Salons 

5.     18-f-|(22—  18)=18  +  2|=£0|  mean  diameter 

4 


MECHANICAL  POWERS, 

1st.       THE    LEVER. 

Page  190. 
As  1  inch  :  20  inches  : :  5  Ib.  :  100  pounds  the  answer 

2d.       THE    WHEEL    AND    AXLE. 

Page  191 
As  60  inches,' =»5  lb-j  :  6  inches  ::  10  Ib.:  1  Ib. 


PROMISCUOUS    QUESTIONS- 

PROMISCUOUS  QUESTIONS. 

Page  192. 
Quest.     1.     2578+2578=5156 

2.  14676—  14f76  =14676—  3669=11007 

3.  1468—  f461  +  5Si;  =  1468—  1042=426 


4.  |  of  100  cents  =33^  cents 
iof  i=j-of  100cts.=162       ' 

Sum  50   cents 

5.  1080-r-45=24  the  number  required 
6. 


7.     Suppose  one  side  of  the  square  to  consist  of  100  men 
Then,  1002  +231  =10231  the  whole  number  of  men 
And    1012—  44=10157  the  whole  of  the  men 


74  error  too  little 

Again,  suppose  the  side  of  the  square  to  have  120  men 
Then,  1202  +  231=-14631  the  whole  number  of  men 
And  1212—  44=14597  the  whole  number  of  men 

34  error  too  little 
Errors  alike. 

(12QX74>-  flOOX34; 
Then,  l ~ — —— ^=5||°  =  137  men  on  one  side 

7*  —••'—  o  i 

Lastly,  (137;2+251,  or  (13SJ2—  44=19000  men 

8.  113X147— 213=16611— 9261=7350 

9.  f  of  t=f=£  then,  as  £  :  1260  d.  ::  I  :  5040  dolls. 

10.  3500—2500=1000  dollars 

Then,  2500  dollars  )  dolls.  C  100  dollars 

8  years   $  1000  £     1  year 
40 


128  PROMISCUOUS    QUESTIONS* 

Page  193. 

210x3 

11.  As  §  :  210  ::  1  :  — — -  =8315  elder  brother's  por. 

Now,  315X5X^=1890 dollars 

12.  (650— 130;x400=520  d.  x4  dolls  =2080  dolls- 

13.  17+8+46+f20x2;=lll  years 

14.  1000— f350+400;=250  dollars  C's  share 

dolls,   dolls.  dolls,    dolls. 

Then,  as  250  :  500  • .  •  as  1  :  2  : :  350  :   700  A  put  in 
And,  as  1:2::  400  dolls.   :  800  dolls.  B's  cloth  is  worth 

15.  |  of  2720=544  dollars  the  captain's  share 

gT'gQ 544 

Then,   — — =2fy56 =13  d.  60  cts.  a  sailor's  share 

lOU 

16.  As  6  :  100  ::  972  dolls.  :  16200  dolls,  her  portion 
And,  as |  :  16200  dollars  ::  1  :  27000  dollars 

Then,  27000-r-3=9000  d.  int.  of  the  father's  estate  1  year 

dolls.        dolls. 
Lastly,  as  6  :  100  ::  9000  :  150000  the  father's  estate 

17.  ( 4ft.  6  inj x(2  ft.  9  in.;  X(3  ft  4  in.;=41  sq.  ft.  S  in. 
18.     1+ 


18 
Then,  as  *V  :  12  ::  1  :  12x18=216  feet 

Page  194. 


19.  r-y  =2l|  yean 

o 

SO.     As  ,y  =1  of  |;  :  1710  :  :  1  :  1^^=3800  dolls. 

21.     622(623J2  =3969X992,25=3938240,25  product 
And  C63xV^—r63  +  62V=198V—  94,5=1890  diff. 


mi.  day     miles  da.  h.  min. 

22.  As  68  f=34x2;  :  1  ::  2000  :  2Jf°=29  9  52{f 


PROMISCUOUS  QUESTIONS,  129 

Page  194. 

Ib.  times        Ib.  C.  qr.    Ib.         ftS^V^ 

23.     6:2::  885(=7  3  17)  :  ^-^==295  times 
M.     Suppose  1,  then  1-ff—f  sum 
And,  as  4  :  Jfe:  20  : 


40 


26.     |-15 


28.    Suppose  the  number  to  be  6 

Then,  6x3  —  8=-10  the  whole  of  his  money 
And,    6x2+3=15  the  whole  of  his  mpney 

5  error 

Again,  suppose  10  to  be  the  number 
Then,  10x3  —  8=22  his  money 
And,    10x2-{-3=23  his  money 

1  error 
Errors  alike, 

_T         5X10—6X1 

Now,  -  -        —  sss^cali  children 

D"  '  '  '1 

29.     Inverse  100  dollars  7  year  C  500  dollars  inverse 
6  dollars  3     1     \  500  .dollars 


Then,  10,0^6  g        ft 

J 


30.    Suppose  the  number  to/be  100 
Then,  1f°  +  15  +  l-0o  +  10=:io8i  the  members 
But,  1081—100=81  error  too  little 
Again,  suppose  200  members  in  all 
Then,  2|0  +  15  +  2§°-f  10*19  If  the  members 
But,  200—191f=85  error  too  much 
Errors  unlike. 

Lastly, 


130  PROMISCUOUS    QUESTIONS 

Page  194. 

SI.  360  deg.  x69£  miles=  25020  miles  round  the  earth 
Then,  as  20  miles  :  1  day  ::  25020  :  252°o20==:  1251  days 
Lastly,  1251  days-r-365|  days=3  years  155?  days 

Page  195. 

32.  100X3|=  325 
150X4£=  675 
204X51=1173 

454  )2 173(4  months  23-||f  days 

da.    w.    da.   w. 

S3.     As  7  :  1  : :  1  :  \  A  can  do  alone  in  one  day 
And, as  12  :  1  ::  l  :  ^  the  part  B  can  do  in  a  day 

Then,  f +TV  =  ""gT-^B  the  part  of  the  work  that  they 
can  do  in  a  day,  working  together 
Lastly,  as  jj  work  :  1  day  : :  1  work  :  1|=:4^  days 

34.     (U057— 1)X400^,40710042265625X400 

1,05—1  ,05 

=3256,80338125=3256  dollars  80  cents  3  milla 

TTio.  dolls,  mo.   d. 
S5.   '  As  12  :  5  ::  4  :  If  and  100+l|=101f 

Then, 

doiis.       doiis.  i  nn  v  ^nn  v  ^ 
As  s|5(-101|)  :  100  ::  700  :       ^ --=688  52  4|f 

01  4.04 
36.    f+^^-^ii  and  !_}!= ft  C's  parts 

7 

Then,  asJJ.:f!40::f  :^ 
8 


.  i  -     * 

And,  asfj-  :  140  ::  $  :    P===3^°  =30545^^  paid 


_     xu 
*RULE.    A=  —  5  —  -  -  .    Here  w=the  annuity  R=* 

lv  —  I 

the  ratio  or  amount  of  £.1,  or  gl,  in  one  year  as  in  com- 
pound interest;  1=  the  time  for  which  the  annuity  is  ia 
arrear.  A=the  amount  of  annuity  at  the  end  of  t  years* 


PROMISCUOUS    QUESTIONS.  131 

Page  195. 

57.  Suppose  the  number  to  be  12. 
Then,   12x4  +  16=64  his  money 
And,  12x6—12=60  his  money 

4  error  too  little 

Again,  Suppose  20  to  be  the  number  of  beggars 
Then,  20X4+16=  -96  his  money 
"And  20X6  —  12=108  his  money 

12  error  too  much 

Errors  unlike. 


Th8D,  =¥-14  beggars 


38.  As  18  :  1  ::  1   :  ^  the  part  B  and  C  can  do  in  a 
day  working  together. 

And,  as  11  :  I  ::  1   :  ^T  the  part  of  the  work  they  can 
do  in  a  day,  all  working 

18  _  i[ 
Now,  VT  —  rV^     1Qo  '  ^TSF  tne  Part  of  the  work  that 

A  can  do  in  a  day 
Lastly,  as  ^  :  1  ::  1  :  1|8  =  28  days  3  hrs.    25f  min* 

39.  20  congress  each  spends  4=5  dollars 
30  merchants  each  spends  y=6      ' 
£4  lawyers  each  spends    |=3      ' 

24  citizens  each  spends    7\=2      < 

16 

d.  dolls. 
dolls.        C  5  :  60  paid  by  the  congress 


40. 


is  16  :   192 

_             „  r  —  .  ~j 
..     )  6  :  72 
)  3  :  36 

(  2  :  24 

merchants 
lawyers 
citizens 

< 

Pagel96.il 

28X20       = 
14X10X2= 

560  square  inches 

280         «         ' 

280=  1  1  acres 

PROMISCUOUS    QUESTIONS. 

Page  196. 

41.    200  acres=S2000  perches=6 
40  perches  =a.    The  rule  is,* 

\X(|2+&)— I  the  short  side,  and^/(°2+&)+|  the  long  side 
Thus,  V^-i  +32000)- 42°=160  the  short  side 
And,  ^/(^   +32000)  +  ^=200  the  longer  side 

42.  Suppose  the  side  of  the  square  meadow  to  be  1 ; 

Then,N/l2  +  l2=v/2=l,41421S6  its  diagonal 
And,     1,4142136— 1=,41421S6  what  the  diagonal   is 
more  than  the  side 

Now  say,  as  ,41421 36  :  1  ::  20  :  48.28425  the  side  of 
the  required  square  in  perches.  Lastly,  square  the  side, 
and  we  get  the  contents  2331, 37  perches=14a.  2r.  llp.f 

L. 

43.  As  18  :  1  ::  24000(=100)  :  2 \ P-°°  =  ^oo  SqUare  ft. 
And,  4o3oo^8g3=:40ioox_i__iooo  feet  perpendicular 

Then, 


,e  of  the  equal  sides; 
consequently  ^</ 19 42619-=  106, 85  &c.  the  sum  required. 

Solution  by  Algebra. 

*  Let  x=  the  short  side,  «=1he  difiference  of  the  sides, 
and  6=the  contents  in  perches 
Then,  (x+a)xx=t\\Q  contents  of  the  ground  in  perches, 

That  is,  x2+ax=b  by  the  question. By  comple- 

2  2 

ting  the  square  we  get  x2-f-a,r-f  ^  =^L  +b;  and  by  fevoiu* 
tiona;-j-|^v/5  +^5  consequently  x^^'L  ~^lL£ 

t  Put  a:=the  side  of  the  square,  and  a=20. 

Then,  fEuclid  1.  47)2x2=(x+a)2=x2+2ax  +x*  bj 
transposition  we  get  x2 — 2ax=a2  and  by  completing  the 
square  #2-2<2#-|-a2=2ft2,  by  evolution  x — «=\/2a2  hence 
a;=cr-{-av/2.  Now  by  restoring  the  value  of  «  we  get  x  the 
side  of  the  square  to  be  20+20^2=20  +  20x1,4142136 
=48,284272  perches  in  the  side  the  same  as  above,  whence 
the  contents  may  be  readily  found. 


PROMISCUOUS    QUESTIONS.  *3 

Page  196. 

Question  43  continued.  Or,  decimally 

As,   18  :  1  ::  24000  :  1333,33  &c.  square  feet 

1  0(T»n     (?  O 

And,    —    — =30,303  &c.  the  perpendicular 

Then,  2  %/442^^S03i-=2v/£854^74-  -=.106,85  feet 

44.  Half  an  acre  80  per.  X30|  =^2420  square  yards 
Then,  2-^L =3081, 23  the  square  of  the  diameter. 

5<  OO-'i 

v/3081,23     55.5 

Z . —  =_  =27.73  yards 

2  2 

45.  114  yards  6  feet=1032  square  feet 
Then,  1032-r-28=SGf  feet 

f  f+         /     <N  ^5  2\'<«  «   .  -    ,    r, 

46.  _ »y«^t1f  inciies 

y 

47.     20  feet^240  inches 

And,  240X11  Xt4=240x|Xf^64§-°  =375  cubic  inches 
Half  a  ton  is  1120  pounds;  also,  50  feet— 600  inches 
And,600xIxi=2V4°°~459|  cubic  inches 

cubic  in.         Ib.  culic  in.  Ib. 

Then,  as  375  :  1120  ::  459s  :  1S72 

lb.         d.  Ib.  pence          L>     ».    «. 

Lastly,  as  1   :  3d  ;:  1372  :  4802=20  0  2 


48.  Here  v/392—-152=V  1  oil— 225 = 

36  faet  standing 
Then,  36+39=75  feet  the  whole  height 

49.  Here  work  backwards 

Thus,  £4x9=216;  ^/2l6— - 6:  Cx^-=8;  82=64;  64— 
4=r60;  6D.-r-5=12;  12 X 8—965  lastly,  96+7—103. 

Page  197. 

50.  Here  (lie  several  portions  of  wine  drawn  off  were 
50,     remains        450 

50X450  .  50X450     4503 

rema 
12 


154  PROMISCUOUS    QUESTIONS. 

Page  197. 

Question  50  continued, 

5QX4502         .     '450*      50  X  450s     450s 
"S          "    " 


5002 
50X4503         .     4503      50X45G34504 


500^    500^ 
50X4504      4504   50X4504  4505 

«         -  - 


_ 

5004        500'    500* 
450*   1845£81  £500000 


Consequently,  500—  295^9o:=204|-|(1rgall«.  of  water* 

*  Analytically.  Put  500=c,  50=</,  5~t,  and  or^what 
wine  remains  after  drawing  /  times 

Then,  c  —  d  expresses  the  quantity  of  wine  left  after  the 
first  drawing;  c  :  c  —  d  ::  d  :*—  -  the  quantity  of  wine 

drawn  out  at  the  second  drawing,  and  c  —  d 


/c  _  ^\2 

\  -  lj_the  quantity  of  wine  left  after  the  second  drawing. 


c—  ,     —  ..        f      . 

Also,  c  :  v-  --  —  ::d;.-  -  r  -  tne    quantity   of   wme 

2 


drawn  out  at  the  third  drawing. 


. 


> 

c         .        c2  c* 

wine  left  after  the  third  drawing;  and  universally 


the  quantity  of 


~^the  ouantity  of  wine  drawn 


c  « 

out  at  the  i  drawing. 


And,         _ 

J—  —  Xi 

c  c  '. 

remainino;  after  the  t  drawing,     Ey  substituting  the  values 
of  c,  d,  ami  t,  in  this  formula,  we  get 


(£00—  50)^450^^  u       f    .      the  game  ag  aboye 

4  * 


PROMISCUOUS    qUESTIONS,  135 

Page  197. 
4X4X8~128~ 

52.     1817,  April  1st.  principal  1864  dollars 

Oct.   15th    1864  doll,  mult  by  197  days=367208 
Paid  225,50 


1818  Jan.  10th    1638,50  X         87       =142549,5 

Paid     188,86 

May  16th    1449,64  X       126       =182654,64 

Paid    585 


Aug.  1st.     864,64  X         77       «=  66$?f,28 

Interest  126,49  ,  . 

6|0)75898]9,42 

Balance  §991,13  '- 

Interest  £126,49 

S6x6ix8£     2047-1 

53.     — -i28"=  15'2-f«-  cords 

days     dolls,  cts,     day 

54.  As  365  :  356    34  ::  1  :  97  cents  6f°  mills 

cts.    doll,     dolls,    cts.         dolls,     cts. 

55.  As  18|  :  1  ::  2564  95|   :  13679  76 

56.  Inverse  8  men  >  days  \    64  men  inverse 

820,50     5    30    {  100  dollars 

25 


30X^X^0000     30X25 


41 


57.     Inverse  34  men  )  months  C  86  men  inverse 
1  bridge  5       6       ^2  bridge 

months 


135  PROMISCUOUS    QUESTIONS* 

Page  197. 

58.  As  4  :  6  :  :  r  :  10^=  10,5  dollars  C  must  pay  v 
A  pays  o  dollars  and  B  7  dollars. 

And,  5  +  7+10£=22i=2-2,5  dollars 

dolls,   dolls,     f   5     :  g    53  S3  3  A  receives 
Lastly,  as  22,5  :  240  :  :  4    7     :  g    74  66f  B        ' 
(,10,5  :  S  112  00    C        * 

Page  198. 

dolls,  cts.  7775.    dolls,  cts.  dells,  cts. 

59.  As  100  :  7     S  ::  8564  20  :  6    25+  tax 

60.  f4  galls.  2  qt.)—  (2  galls.  2  qt.  1  pt.)  =  l  gall.  3  qt 
1  pt.=15  pints  gained,  or  remaining  in  the  vessel  at  the 
end*ftf  one  hour. 

And,  84£  gallons  is  676  pints 
Then,  as  15  pts.  :  1  hr.  :  :  676  pints  :  45  hours  '4  minutes 

P.      ft.       ft.  *q.J*t 

61.  One  acre!60xl6,5xi6,5=160x272|=4356a 

Then,  43560—36=1210  feet=73§  perches 

50  men    bushels  C  40  men 
12     £  90  days 


63.      Bor        IX  6=  6  cents 
Women  3  X  8=24     ; 
Men       6X16=96     c 


126  cents 

f  6  :      90  cents  the  boys  get 

Then,  as  Or:  -{.24  :     S60  cents  the  women  get 

[96  :  1440  cents  the  men  get 

Consequently,  90—  6=15  boys     "1 

-H  8=45  women  t 

And  1440-4-  16=90  men      J 

G4.     7X4J—  8x3  =  SlJ—  24=7i  miles  apart  in  one  day 
1  day  :  7\  mi.  :  :  13  days  :  97d  miles  apart 


PROMISCUOUS    QUESTIONS.  137 

Page  198. 

65.     962X6=5772        1 635  XS£ =5722^ 
320  4081 


1282X6=7692        12261x5=6131,25 


A   13464  B    11853,75 

A  13464 
B   11853,75 


25317,75    Then, 

dolls,  cts.  ms. 

dolls,  ds.    C  13464         :   258,79,5^14 
As  25317, 75:  486  64::  £  11853,75   :  227,84,4^|ff 

Proof  g486,64 
52803X23 


66.      Here    2500x6x5280X3 
44 

1056 


75 

25 
=4956^-  hours—  206  days  12  hours  9  min.  36  seconds 

67.    45  j  ft—  9|  in.  =546  in.—9j  in.=536^  inches 
And  9|  in.  —  6|  in.  =2|  inches  gained  in  a  day 
Now,  536^~-2j  =  1%73XT4T=a^6=195T1r  days 

That  is,  it  will  be  within  9|  inches  of  the  top  in  195^ 
dayi.    Consequently  it  will  get  to  the  top  in  196^  days. 


tree. 


PROMISCUOUS    QUESTONS* 

Page  198. 

69.     Suppose  200  dolls.     Again,  Suppose  300  dollare 
94  94 

106  206 

ioM  05=  53  lent  £  of  206=  103    lent 

J  of  106=  21  \  clothes  £  of  206=  4l£  clothes 

94  debts  94    debts 


£  sum  gone  g238£  sum  gone 

T'oGf200=lSO  •&  of  300=270 

1  1  £  error  too  much         error  5  1|  too  much 
Errors  alike, 

200  X  3  1  1—300  X  1  1$     6360—3540 

Then>  —  3ij-iii    —  —  «»—  1 

Page  199. 


Then,  asTV  :  84  ::  1  :  84X12=1008 

71.    Here  3—  •£  :  4ft.  ::  3  :  12  feet  the  semidiameter 
of  the  circle  that  the  outer  wheel  makes.     Consequently 
12x2=24  feet  the  diameter  of  the  outer  circle. 
And,  24  —  4x2=16  feet  the  diameter  of  the  inner  circle 
Lastly,  24X3,1416=75,3984  feetbj  the  outer  wheel 
And,  13x3,1416—50,2653  feet  by  the  inner  wheel 


PROMISCUOUS    QUESTIONS. 

Page  199. 

72.    Here*  1|5=one  of  the  equal  sides,  1|5=half  the 
sum  of  the  sides 

^75     o^n 
And  '|5 — l$5=- — =1|5  one  of  the  equal  rem. 

Then,  V/ 

125*    .       15625    .        15625X1,7320508 
"6T"^/3==~^rx/3  ~T6~ 

751,75816  square  feet. 

Or,  by  RULE  n,*  thus;  125—3=  1|5  is  one  of  the  equal 
sides  of  the  triangle,  and  because  the  triangle  is  equilate- 
ral, and  consequently  equiangular,  the  included  angle  of 
any  two  sides  is  60°,  whose  natural  sine  is  ,8660254 

Hence,  1|5X1|5X,866|254=751,7581596  square  feet 

*  Here  data  is  given  to  find  the  sides. 

The  rule  for  solving  this  question,  and  all  of  a  similar 
kind,  is  by  mensuration,  as  follows: 

1.  From  half  the  sum  of  three  sides  subtract  each 
side  severally. 

2.  Multiply  the  half  sum  and  the  three  remainders  con- 
tinually together,  and  the  square  root  of  the  product  will 
be  the  area  required. 

RULE.  ii.  Any  two  sides  of  a  triangle  being  multiplied 
together,  and  the  product  again  by  half  the  natural  sine  of 
there  included  angle,  will  give  the  area  of  the  triangle. 

That  is,  AC  xCBx  natural  sine  of  the  angle  C= twice  area. 


JOHNSTON  &  STOCKTON, 


Have  just  published  the  Sixth  Edition  of  the 


BY  THE  REV.  JOSEPH  STOCKTON. 
A£S0,  HAVE  CONSTANTLY  ON  HAND, 


JFestern  Spelling  Book 
United  States    do. 
Juvenile  do. 

CobVt  do. 

English  Redder 
Introduction  to  do, 
Murray's  large  Sf  small  Grammav 
Kirkham's  do* 

Pocket  Bibles 
School    do. 
Do.    Testaments 
Pocket        do. 
Watts'  Psalms  and  Hymm 


Olney's  Geography  and  Adas 
IVoodbridge^s  do.  do. 

Smiley'  s          do.  do, 

Cummings*      do.  do, 

David's  Psalms 
Juvenile  Reader 


New  England  do. 
Broien's  Catechism 
Larger  Catechism 
Catholic  Catechism     ' 
Mag.  and  Com.  Almanatt 
German  do. 


In  Press,  the  Sixth  Edition  of  the 

BEAUTIES  OF  HARMONY. 

JBF  FREEMAN  LEWIS: 

And  the  Second  Edition  of  the 

PENNSYLVANIA  HARMONIST; 

AND   REVISED. 


J.  &  S.  continue  to  inanuikcttire,  and 
hare  for  sale,  all  kinds  of  Writing, 
Printing  and  Wrapping 


Which  they  will  exchange  for  Country 
Produce  generally. 

Always  on  hand,   an  assortment  of 
handsome  WALL  PAPERS,  for  rooms, 
window-blinds,  $c> 


JGHIVSTOIf  &  STOCKTON 

Have  generally  on  hand  a  good  stock  of  Books;  and  are  enabled 
to  furnish,  at  a  short  notice,  every  thing  in  their  line  of  business. 

Orders  from  Booksellers,  Country  Merchants,  and  others,  punctu- 
ally attended  to.  . 


ADAMS'  Geography  and  Atlas 
Ainsworth's    Latin    Dictionary, 

8ro 

Adams'  Roman  Antiquities,  8vo 
Almost  Christian,  by  Mead,  I2mo 
Abbot,  by  the  Author  of  Waver- 

ley,  2  vols.  ]2mo 
.  Arthur  Mervyn,  a  Novel,  2  vols. 

12rao 

Arithmetical  Table  Books 
Alleine  on  the  Promises,  18mo 
Adventures  of  H;ijji  Baba  in  Eng- 
land, 2  vols.  I2mo 
American  Common  Place  Book, 

18mo 

American  Chesterfield,  18mo 
Alexander's  Evidences  of  Chris- 
tianity, 18mo,  new  and  cheap 
edition 
Anne  of   Gierstein,    fcy    Walter 

Scott,  2  vols.  12mo 
American  Popular  Lessons,  18mo 
Adventures  of  a  King's  Page,  2 

vols-  ISrao 

Tht  American  Guide,  comprising 
the  Declaration  of  Independ- 
ence, the  Articles  of  Confeder- 
ation, the  Constitution  of  the 
United  States,  and  the  Consti- 
tutions of  the  several  States 
comprising  the  Union.  1  vol. 
18mo 
Bancroft  on  Permanent  Colours, 

2  vols.  8vo 
Brown'g  History  of  Missions;  2 

vols.  8ro 

Bennet's  Letters  to  a  Young  La- 
dy, 18mo 
Beauties  of   Scott   and   Moore, 

18mo 
Beauties  of  Byron.  18mo 

Do.     ©f  Kirke  White,  18mo 
Book  of  Common  Prayer,  with 
the  New  Ilyms,  in  various  sizes 
and  bindings 

Butler's  Feasts  and  Fasts  of  the 
Catholic  Church,  8vo.  Dublin 
Bbiton1*  Fourfold  State,  12mo 
Bossuet's  Universal  Hist.  12mo 
Benjamin'*  Architecture,  4to. 


Bennet's  System  of  Bookkeep- 
ing, 8vo 

Boston's  ('Rev.  Thomas)  View  of 
the  Covenants  of  Grace  and  the 
Covenant  of  Works,  from  tho 
Sacred  Records,  2  vols.  12mo 

Brown's  Christian  Journal,  12mo 

Baker's  translation  of  Livy,  6 
vols.  8vo 

Blair's  Lectures  on  Rhetoric,  a- 
bridged,  with  Questions  to  each 
Chapter,  18mo 

Blair's  Grammar  of  Chemistry 

Buckminster's  Sermons,  new  le- 
ries,  8vo 

Buck's  Theological  Dictionary, 
8vo 

Buchan's  Domestic  Medicine, 
with  the  Advice  to  Mothers,  8vo 

Brown's  Body  of  Divinity,  8vo 

Brown's  Concordance,  neat  edi- 
tion for  the  pocket 

Burnet's  History  of  the  Reforma- 
tion, 6  vols.  18mo,  London  ed~ 

Brown's  Philosophy  of  the  Hu- 
man Mind,  2  vols,  8vo 

Butler's  Lives  of  the  Saints,  5- 
vols.  8vo 

Biitterworth's  Concordance,  8vo 

Brownlee  on  Quakerism,  8vo 

Bible,  with  Scott's  References, 
large  octavo 

Benedict's  History  of  the  Bap- 
tists, ]2mo 

Burns'  Letters,  18mo 

Babylon  the  Great,  2  vols.  12mo> 

Baxter's  Call,  ISmo 

Brown's  Catechism,  small 
Do.     Two  Catechisms,  18m» 

Bonnycastle's  Algebra,  12  mo 

Brown's  Select  Remains,  12mo 

Boswell's  Life  of  Johnson,  5  vols* 
12mo 

Blair's  Grammar  of  Natural  Phi- 
losophy, 18mo 

Chalmers'  Evidences,  12mo 

Cooke's  Voyages,  complete,  t 
vols.  8vo 

Clarke  on  the  Promiaei, 

Cicero  Delphini.  8vo 


Conversations  on  Chemistry, 
Cummings'  Western  Tour,  12mo 
Cicero  de  Officiis,  12  mo 
Cicero  de  Oratore,  J2mo 
Connecticut  Forty  years  since 
Chap  Books,  per  dozen  or  gross 
Charlotte  Temple,  18mo 
Children  of  the  Abbey,  2  vols 
Conversations  on  N.  Philosophy 
Catechism  of  Nature,  18mo 
Caesar  Delphini,  8vo 
Chronicles  of  the  Canongate,  by 
the  Author  of  Waverley,  2  vols 
Carey's  Political  Essays,  8vo 
Cary's  (Mrs.  Virginia)  Letters  on 
Female  Character,  addressed  to 
a  young  Lady  on  the  death  of 
her  Mother,  12mo 
Crook  in  the  Lot,  by  the  Rev. 

Thomas  Boston 
Dick's  Philosophy  of  a  Future 

Slate,  12mo 

Dick's  Christian  Philosopher 
Do.  Philosophy  of  Religion 
David's  Psalms,  pocket  edition 
Do.      with  Brown's  Notes 
Doddridge's  Rise  and  Progress 
Domestic  Cookery,  by  an  Amer- 
ican Lady,  18mo 
Edwards  on  Religious  Affections. 

abridged,  18mo 

Evening  Entertainments;  or  De- 
lineations of  the  manners  and 
Customs    of  various   Nations. 
By  J.  B.  Depping.    1  vol 
Faber's  (Rev.  G.  S.)  Difficulties 

of  Romanism,  12mo 
Family  Bible,   4to,  with    plates, 

Concordance  or  Psalms 
Flavel's  Token  for  Mourners 
Family  Receipt  Book,  12mo 
Franklin's  Life  and  essays,  18mo 
Gummere's  Surveying,  8vo 
Gibson's  do        do 

Goldsmith's  Mariners   and    Cus- 
toms, 2  vols  12mo 
Grimshaw's  United  States,  12mo 
Do.  England  do 

Do.  Rome  do 

Greek  Testaments,  I2mo 
Grimshaw's  History  of  France, 
from  the    Foundation    to   the 
death  of  Louis  XVI.,  12mo. 
Grimshaw's    Life   of   Napoleon, 
with   the     History    of  France 
from  the  Death  of  Louis  XVI. 
to  the  Year  182 J ,  12uto 


Henry's   Commentary,  6   Larg« 

Super  Royal  octavo  vols 
Home's  Introduction  to  the  Stu- 
dy of  the  Scriptures,  4  vols  - 
Heaven  &  Hell,  by  Swedenborg 
Harinam's  Pulpit  Assistant,  New 

edition,  2  vols.  8vo 
Hale's  History  of  the  U.  States, 

an  excellent  School  Bo«k 
Hist,  of  the  late  War,  in  German 
Hay  on  Colds,  18mo 
Henry's    Communicant's    Com- 
panion, 12mo 

Hervey's  Meditations,  18mo 
Hawney's  Mensuration,  12mo 
Horace  Delphini,  8vo 
History  of  the  Indian  Wars,  by 

Trumbull,  8vo 
Josephus,  4  vols.  8vo 
Juvenile  Reader,  ]8mo 
Life  of  Paul  Jones,  by  Sherbumc 
Life  of  Jackson,  by  W.  H.  Eaton 
Life  of  Harriet  Newel,  18mo 
L<?usden's  Greek  and  Latin  Tes- 
tament, 18mo 

Life  of  Washington,  by  Weems 
Life  of  Marion,  by  Weems,  ]2mo 
Life  of  Maj.  Gen.  Putman,  12mo 
Laws'  Serious  Call,  12rao 
Laws'  Address  to  the  Clergy 
Lafayette's  Journey  in  America, 
by  Le  Vasseur,  his  sec'y.    2  v 
Life  of  Luther,  by  Alex.  Bower. 
With  a  full  length  Portrait,  1  v 
Life  of  Calvin.     By  John  Mac- 
kenzie.    With  a  portrait.     Se- 
cond American  edition.    1  vol. 
Millwright's  Guide,  by  O.  Evans 
Mason's  Farriery.  12mo 
Methodist  Hymn  Book,  18mo 
Mackenzie's  5,000  Receipts 
Mills  on  Cattle.  12  mo 
Memoirs  of  Lafayette,  12  mo 
Nicholson's  Farmer's  Assistant 
New  Universal  Letter  Writer,  IT 
Oxven  on  the  Holy  Spirit,  abridg- 
ed by  Burder,  12mo 
Paradise  Lost,  plain  and  elegant 

bindings 
Pilgrim's  Progress,  in  plain  and 

handsome  bindings 
Pocket  Bible,  plain  and  elegant 

bindings 

Pocket  Testaments  de.        do. 
Paley's  Evidences  of  Christinnity 
Presbyterian  Confession  of  Faith 
Ready  Calculator,  8vo 


JOHNSTON  &  STOCKTON 

continue  to  execute 


VIZ. 


BOOKS,                            |             HORSE-BILLS, 

PAMPHLETS,               I 

g          CARDS, 

CIRCULARS, 

%              CHECKS, 

HAND-BILLS, 

NOTICES,    &c. 

Z*T  THE  ATEATEST  STYLE. 

BOOKS  BOITMB, 

In  a  handsome  and  substantial  manner. 


BLANK  BOOKS, 

Of  every  description,  made  of  good  paper*  durably  bound, 
and  ruled  to  any  given  pattern. 

OIiD 


English,  Dutch  and  Opaque  Quills — Black  and  Red 
Ink -Powder  and  Ink — Wedge  wood,  Lead  and  Glass  Ink- 
stands— Pocket  ditto — Sand  Boxes  and  Sand — Slates  and 
Pencils — Paint  Boxes — Pocket  Books — Best  quality  Pen- 
knives—Lead Pencils — Crayons — India  Ink— Red  and 
Black  Sealing  Wax — Wafers— Mathematical  Instruments 
— Drawing  Paper,  &c. 


A  GENERAL  ASSORTMENT  OF 

CLASSICAL  SCHOOL  BOOKS, 


Jl  great  variety  of  Children's  Toy  Books. 
Magistrate's  BlanJis  for  Sale. 


YA  02399 


m 

ma 


